Helix Theory

by A. L. Gorodentsev^{1} and S. A. Kuleshov^{2}

December 2001

Abstract.

This is a detailed review of the helix theory, which describes exceptional sheaves and exceptional bases for the derived categories of coherent sheaves on Fano varieties. We explain systematically all basic ideas and constructions concerned with exceptional objects. Projective spaces and Del Pezzo surfaces are considered especially extensive. Some arithmetical interactions with the mirror symmetry phenomenon are discussed as well. This paper also may be considered as a necessary complement to the book [HuLe], which completely

ignores rich structures staying behind the zero-dimensional moduli spaces.

Contents

Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 §1 Instead of introduction: non symmetric orthogonal linear algebra . . . . . . . 2 §2 Semiorthogonal decompositions in linear triangulated categories . . . . . . . . 8 §3 Helix theory on ℙ_{n} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 §4 Helix theory on Del Pezzo surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 §5 Del Pezzo surfaces: stability of rigid sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 §6 Del Pezzo surfaces: transitivity of mutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 §7 Helices and Riemann - Roch arithmetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

1Institute of Theoretical and Experimental Physics (Moscow) & Independent University of Moscow 2Zhukovsky Airforces Academy (Moscow) & Independent University of Moscow

§1. Instead of introduction: non symmetric orthogonal linear algebra.

1.1. Unimodular Mukai lattices. An unimodular Mukai lattice is a free ℤ-module M ≅ ℤ^{n}, of finite rank, equipped with a bilinear form M × M ⟶ ℤ : v, w ⟼ ⟨ v , w ⟩, which may be neither symmetric nor skew symmetric but has to be unimodular, i. e. produce ℤ-module isomorphism

M ^{m↦⟨ m , ∗ ⟩} ^{}Hom_{ℤ}(M, ℤ) .

An integer base {e_{1}, e_{2}, …, e_{n}} ⊂ M is called exceptional or semiorthonormal, if its Gram matrix χ_{ij} =^{def} ⟨ e_{i} , e_{j} ⟩ is upper triangular with units on the main diagonal, i. e. ⟨ e_{i} , e_{j} ⟩ = 0 for all i > j and ⟨ e_{i} , e_{i} ⟩ = 1 for all i. In general, an unimodular Mukai lattice does not have a semiorthonormal base at all: the unimodularity just means that det ( χ_{ij}) = ±1 for any integer base of M (see n^{∘} 1.4.4 for general facts about non symmetric bilinear forms).

However, unimodular Mukai lattices equipped with semiorthonormal bases appear in the following two well known ‘mirror symmetric’ theories. The first is the Picard - Lefschetz theory, where M is a lattice of vanishing cycles in the middle cohomologies of a smooth level hypersurface for a holomorphic function near an isolated critical point. This lattice comes with Seifert’s (non symmetric) form and its exceptional bases are usually called ‘distinguished’ (see [AVG]). The second theory deals with algebraic coherent sheaves on a Fano variety X. Here M = K_{0}(X) is the Grothendieck ring equipped with the Euler form

⟨ E , F ⟩ = ∑(−1)^{ν} dim Ext^{ν}(E, F)

and semiorthonormal bases are constructed from exceptional objects of the derived category of coherent sheaves � _{�oh}^{b} (X). In the present lectures we discuss this coherent theory (but at the last paragraph we make few remarks towards a bypass to Picard - Lefschetz).

1.2. What is the helix theory. The derived category� _{�oh}^{b} (X) may be considered as a kind of ‘module’ over a kind of ‘ring’ whose role is played by the triangulated category of graded vector spaces with direct sums and tensor products as the ring operations. Grothendieck’s K_{0} functor transforms this intuitive module structure to the ordinary one on the Mukai lattice. The helix theory lifts the basic concepts of orthogonal linear algebra (duality, orthogonal decompositions, e. t. c.) as well as some special effects (like the braid group action on the semiorthonormal bases) from the level of K_{0}(X) onto the level of� _{�oh}^{b} (X). The main difficulty here is that we have to represent all linear combinations used to define/compute the sings in K_{0}(X) by some actual canonical morphisms in � _{�oh}^{b} (X). Fortunately, it is possible almost everywhere as we will see in the next paragraph.

§ 1. Instead of introduction: non symmetric orthogonal linear algebra. 3

1.3. The paper is organized as follows. We finish this introduction by recalling some basic constructions from the (non symmetric) orthogonal linear algebra. In §2 we discuss the helices and orthogonal decomposition technique in general linear triangulated category setup. Then we will consider particular versions of the helix theory: on projective spaces in §3 and on Del Pezzo surfaces in §4. In this connection, in §4, §5, we discuss also the geometry of rigid sheaves and some concepts of ‘stability’. §6 is devoted to transitivity of the braid group action on exceptional collections of sheaves over Del Pezzo surfaces. In the last §7 we make few remarks on interactions between the Helix theory and the Diophantine approximations on the one side and rigid local systems on ℙ_{1} \ { some points } on the other side.

1.4. Linear algebra draught. Before we begin an abstract rubbish on orthogonal decompositions in� _{�oh}^{b} (X), let us outline some basic concepts of the integer non symmetric orthogonal linear algebra on the level of Mukai lattices.

1.4.1. Mutations and orthogonal projectors. Let M be a Mukai lattice and e ∈ M have

⟨ e , e ⟩ = 1. We write

⟨^{⊥}e⟩ =^{def} {m ∈ M | ⟨ m , e ⟩ = 0} and ⟨e⟩^{⊥} =^{def} {m ∈ M | ⟨ e , m ⟩ = 0}

for its left and right orthogonals. The maps

\begin{aligned}M ^{L_{e}}^{}⟨e⟩^{⊥} & : m ⟼ ⟨ e , m ⟩ · e − m \\ M ^{R_{e}}^{}⟨^{⊥}e⟩ & : m ⟼ ⟨ m , e ⟩ · e − m \end{aligned}  (1-1)

are called left and right mutations w. r. t. e. Up to the sign, they coincide with the left orthogonal projection onto ⟨e⟩^{⊥} and the right orthogonal projection onto ⟨^{⊥}e⟩, that is, ∀ m∈ M we have

⟨ m , x ⟩ = − ⟨ L_{e}m , x ⟩ ∀ x∈⟨e⟩^{⊥} and ⟨ x , m ⟩ = − ⟨ x , R_{e}m ⟩ ∀ x∈ ⟨^{⊥}e⟩ . (1-2)

We call L_{e} and R_{e} left and right mutations with respect to e.

1.4.2. Duality and orthogonal decomposition. Let {e_{0}, e_{1}, …, e_{n}} ⊂ M be an semior-

thonormal base. Its left and right dual semiorthonormal bases {^{∨}e_{i}} and {e_{i}^{∨}} are defined by the orthogonality relations:

(−1)^{i} ⟨^{∨}e_{i} , e_{n−j} ⟩ = (−1)^{n−j} ⟨ e_{n−j} , e_{i}^{∨} ⟩ = δ_{ij} .  (1-3)

By (1-1) and (1-2), they can be constructed using consequent mutations as follows

e_{i}^{∨} = L_{e_{0}}L_{e_{1}} … L_{e_{n−i−1}} e_{n−i}
∨e_{i} = R_{e_{n}}R_{e_{n−1}} … R_{e_{n−i+1}} e_{n−i}

For any m = ∑ x_{ν}e_{ν} ∈ M the coordinates x_{ν} can be computed as

x_{ν} = (−1)^{n−ν} ⟨^{∨}e_{n−ν} , m ⟩ = (−1)^{ν} ⟨ m , e_{n−ν}^{∨} ⟩ .

In §2 this formula will turn to a representation of any object m ∈ Ob� _{�oh}^{b} (X) as a canonical contraction of some functorial complex of exceptional objects; this gives natural spectral sequences for computing the values of cohomology functors at m.

If E ⊂ M is a sublattice generated by some semiorthonormal collection {e_{0}, e_{1}, …, e_{n}}, then there are left and right orthogonal projections M ^{}E as well as the left orthogonal projection M ^{L_{E}}^{}E^{⊥} and the right orthogonal projection M ^{R_{E}}^{}^{⊥}E. The last two generalize (1-1) and take

\begin{aligned}L_{E} & : m ⟼ m − ∑_{ν}(−1)^{n−ν} ⟨^{∨}e_{n−ν} , m ⟩ · e_{ν} \\ R_{E} & : m ⟼ m − ∑_{ν}(−1)^{ν} ⟨ m , e_{n−ν}^{∨} ⟩ · e_{ν} \end{aligned}  (1-4)

where {^{∨}e_{ν}} and {e_{ν}^{∨}} are left and right dual to {e_{ν}} bases for E.

1.4.3. Canonical Serre operator. There exist a canonical linear automorphism M ^{κ}^{}M such that

⟨ m_{1} , m_{2} ⟩ = ⟨ m_{2} , κm_{1} ⟩ ∀ m_{1}, m_{2} ∈ M

Its matrix is expressed through the Gram matrix as κ = χ^{−1}χ^{t}. If {e_{0}, e_{1}, …, e_{n}} and {^{∨}e_{0},^{∨}e_{1}, …,^{∨}e_{n}} be a pair of dual exceptional bases for M, then

κ : m ⟼ ∑_{ν}(−1)^{ν} ⟨ m , ^{∨}e_{n−ν} ⟩ · e_{ν}  (1-5)

Indeed, the orthogonality relations (1-3) show that if {ε_{0}^{∨}, ε_{1}^{∨}, …, ε_{n}^{∨}} is the right dual base to {ε_{0}, ε_{1}, …, ε_{n}}, then {κε_{0}, κε_{1}, …, κε_{n}} is the right dual base to {ε_{0}^{∨}, ε_{1}^{∨}, …, ε_{n}^{∨}}, that is, writing κε_{ν} = e_{ν} we get

m = ∑_{ν}(−1)^{ν} ⟨ m , ε_{n−ν}^{∨} ⟩ · ε_{ν} ^{κ}^{}∑_{ν}(−1)^{ν} ⟨ m , ε_{n−ν}^{∨} ⟩ · κε_{ν} =

= ∑_{ν}(−1)^{ν} ⟨ m , ^{∨}e_{n−ν} ⟩ · e_{ν}

In the categorical setup κ is realized up to a sign by the Serre - Verdier dualizing functor.

1.4.4. Digression: non degenerated bilinear forms over ℂ. From the standpoint of the

invariant theory, the map

χ ⟼ κ = χ^{−1}χ^{t} ,

which sends bilinear forms to linear operators, is clearly GL(M)-invariant. If we extend M to a vector space \overline{M} = ℂ ⊗_{ℤ}M over ℂ, then one can prove that this map separates the

orbits, i. e. two forms are equivalent iff the corresponding canonical operators are conjugate. Moreover, its image may be explicitly described, say, in terms of all Jordan normal forms that are possible for κ (see [Go5] for more details and further references). This leads to the following classification of vector spaces V equipped with non degenerated bilinear forms up to isometries^{1}. We call V decomposable, if it splits into biorthogonal direct sum of proper subspaces:

V = V_{1} ⊕ V_{2} , where ⟨ V_{1} , V_{2} ⟩ = ⟨ V_{2} , V_{1} ⟩ = 0 , V_{1} ≠ 0 , V_{2} ≠ 0 .

Clearly, each V is a biorthogonal direct sum of indecomposable spaces. Indecomposable space is isometric either to the space W_{k}(λ) of dimension 2k with the form

(\begin{matrix}0 & I \\ I_{λ} & 0 \end{matrix}) , where I = (\begin{matrix}0 & & & & 1 \\ & . & . & . \\ 1 & & & & 0 \end{matrix}) , I_{λ} = (\begin{matrix}0 & & & 1 \\ & & 1 & λ \\ & ... & ... \\ 1 & λ & & 0 \end{matrix}) , λ ≠ ±1

1i. e. form preserving linear isomorphisms

(non isometric to each other for different λ) or to the space U_{n} of dimension n with the form

(\begin{matrix}& & & & 1 \\ & 0 & & −1 & 1 \\ & & 1 & −1 \\ & ... & 1 \\ ... & ... & & 0 \end{matrix}) .

1.5. Braid group action. Let the standard generators g_{1}, g_{2}, …, g_{n−1} of B_{n} act on the semiorthonormal collections {e_{1}, e_{2}, …, e_{n}} ⊂ M as

g_{i} : \{ \begin{matrix}e_{ν} & ⟼ e_{ν} for ν ≠ i , (i + 1) \\ e_{i} & ⟼ e_{i+1} − ⟨ e_{i} , e_{i+1} ⟩ e_{i} \\ e_{i+1} & ⟼ e_{i} \end{matrix}

In other words, we transpose the pair of consequent vectors (e_{i}, e_{i+1}) and then restore the broken semiorthonormality by the standard orthogonalization procedure, which can go in two directions:

[figure: (e_{i}, e_{i+1}); (e_{i+1}, e_{i}); ( e_{i+1} − ⟨ e_{i} , e_{i+1} ⟩ e_{i} , e_{i} ); g_{i}; ( e_{i+1} , e_{i} − ⟨ e_{i} , e_{i+1} ⟩ e_{i+1} ); g_{i}^{−1}] (1-6)

If we leave e_{i} and change e_{i+1} by e_{i+1} − ⟨ e_{i} , e_{i+1} ⟩ e_{i}, then we get g_{i} written above. If leave e_{i+1} and replace e_{i} by e_{i} −⟨ e_{i} , e_{i+0} ⟩ e_{i+1}, then we get the inverse transformation g_{i}^{−1} actually. A direct computation shows that the braid group relations

g_{i}g_{j} = g_{j}g_{i} for |i − j| > 1
g_{i}g_{i+1}g_{i} = g_{i+1}g_{i}g_{i+1}

hold for this action. So, in order to describe all the semiorthonormal bases of M we have to study the braid group orbits. An other natural group acting on M is the isometry group Isom(M), of all form preserving linear automorphisms.

1.5.1. CONJECTURE. If \overline{M} = ℂ ⊗_{ℤ}M is isometric to U_{n} (see n^{∘} 1.4.4), then, up to the

signs of basic vectors, a semidirect product Isom(M) ⋌ B_{n} acts transitively on the set of all semiorthonormal bases in M.

This conjecture is not proved yet but it is verified for a number of algebraic geometric examples, where M = K_{0}(X). It is closely connected with classical problems from the geometry of numbers. Postponing this subject to the last paragraph, we mention just two basic examples here.

1.6. Example: K_{0}(ℙ_{2}) and the Markoff numbers. As we will see in n^{∘} 7.2, there exists a unique (up to isometric isomorphism over ℤ) rank 3 lattice M ≅ ℤ^{3} that is indecomposable over ℂ and admits semiorthonormal base. This is M = K_{0}(ℙ_{2}). Moreover, elements of any semiorthonormal Gram matrix

( ⟨ e_{i} , e_{j} ⟩) = (\begin{matrix}1 & x & y \\ 0 & 1 & z \\ 0 & 0 & 1 \end{matrix})

satisfy the triple Markoff equation

x^{2} + y^{2} + z^{2} = xyz ,  (1-7)

(see details in [Go5], [GoRu] and n^{∘} 7.2 below). So, there is a 1 – 1 correspondence between the isometry classes of semiorthonormal bases and triples of the the triple Markoff numbers (i. e. the the solutions of (1-7)). Up to the signs, all the solutions of (1-7) are obtained from the ‘initial’ one (3, 3, 3) by natural quadratic transformations

(x, xz − y, z)

(x, y, z)^{}

[figure: (yz − x, y, z); (x, y, xy − z)]

predicted by the Vieta theorem (see [Ca]). The initial triple (3, 3, 3) corresponds to ‘the exceptional base’ {� , � (−1) , � (1) } for K_{0}(ℙ_{2}) and the above transformations may be organized into B_{3} - action on the triples of the solutions for (1-7) in such a way that the correspondence in question would B_{3} - equivariant (see n^{∘} 7.2 below). So, up to signs of basic vectors, B_{3} acts transitively on the isometry classes of exceptional bases of M.

Originally, the Markoff numbers appear more than 100 years ago in view of the following phenomenon. For any totally real^{1} quadratic form q(x, y) a homogeneous minimum is defined as

μ(q) = \frac{1}{\sqrt{|det(q)|}} min_{(x,y)∈ℤ^{2}\(0,0)}| q(x, y) | .

Then, there exists a unique (up to SL_{2}(ℤ)-action) form q_{1} whose μ(q_{1}) is maximal among all totally real q’s. Outside the SL_{2}(ℤ)-orbit of q_{1} there is again a unique up to SL_{2}(ℤ)-action form q_{2} with maximal μ(q_{2}) and so on. The sequence μ(q_{1}), μ(q_{2}), …, of these isolated minima, is called the Markoff spectrum and the corresponding forms q_{i} are known as the Markoff forms. The Markoff forms are in 1 – 1 correspondence with the solutions of the Markoff equation, which appears initially as a tool for building the Markoff forms (see [Ma]).

It turns out that under the above correspondence between semiorthonormal bases of M = K_{0}(ℙ_{2}) and triple Markoff numbers the Markoff forms appear naturally as the restrictions of the quadratic form q(v) = ⟨ v , v ⟩ from M onto rank 2 sublattices ⟨e⟩^{⊥} for all e ∈ M that can be included in a semiorthonormal base of M.

1this means that q is an (irreducible over ℚ) product of two linear forms with real coefficients

1.7. Example: semiorthonormal basises of rank two Mukai lattices. Let M have a semiorthonormal base {e_{0}, e_{1}} with the Gram matrix

(\begin{matrix}1 & h \\ 0 & 1 \end{matrix}) , where h > 0 .  (1-8)

Consider two operations L, R taking a semiorthonormal pair {u, v} to

L({u, v}) = {L_{v}(u), v} , where L_{v}(u) = ⟨ u , v ⟩ v − u
R({u, v}) = {u, R_{u}(v)} , where R_{u}(v) = ⟨ u , v ⟩ u − v

They are inverse to each other, preserve the Gram matrix, and up to a sign coincide with the left and right mutations (1-6), which define an action of the braid group B_{2} = ℤ on semiorthonormal bases of M. Applying R and L to the initial base {e_{0}, e_{1}}, we get a series of unit vectors {e_{i}}_{i∈ℤ} such that R({e_{i−1}, e_{i}}) = {e_{i}, e_{i+1}} = L({e_{i+1}, e_{i+2}}), or equivalently, e_{i−1} + e_{i+1} = he_{i} ∀ i∈ℤ .

1.7.1. PROPOSITION. All semiorthonormal bases of M, up to a sign, are exhausted by {±e_{i}, ±e_{i+1}}. If h ≥ 2, then e_{i} ≠ e_{j} for i ≠ j and each e ∈ M such that ⟨ e , e ⟩ = 1 coincides with some ±e_{i}, i ∈ ℤ; the isometry group^{1} Isom(M) ≅ ℤ_{2} × ℤ. If h = 1, then there are exactly 6 vectors e ∈ M such that ⟨ e , e ⟩ = 1: up to a sign, they are e_{0} = u, e_{1} = v and e_{2} = v − u and the isometry group is Isom(M) ≅ (ℤ/2ℤ) × (ℤ/3ℤ), i. e. all semiorthonormal bases of M, up to a sign, are exhausted by

{e_{0}, e_{1}} [figure: R; L] {e_{1}, e_{2}} [figure: R; L] {e_{2}, e_{0}} [figure: R; L] {e_{0}, e_{1}} ;

In the both cases, Isom(M) is generated by −Id and ϱ : {e_{0}, e_{1}} ⟼ {e_{1}, e_{2}}. Proof. Identify M with the integer sublattice ℤ^{2} ⊂ ℝ^{2} spanned by the cartesian base {e_{0}, e_{1}}. Then v = xe_{0} + ye_{1} ∈ M with ⟨ v , v ⟩ = 1 give all integer solutions of the equation x^{2} + hxy + y^{2} = 1. If h = 1, then all x, y ∈ ℤ such that

x^{2} − xy + y^{2} = (x − y)^{2} + xy = 1

are (up to a sign) (1, 0), (0, 1), and (1, 1). This implies the second part of the proposition. Now, let h ≥ 2. The operator ϱ : {e_{0}, e_{1}} ⟼ {e_{1}, e_{2}} has the matrix

(\begin{matrix}0 & −1 \\ 1 & h \end{matrix}) ∈ SL_{2}(ℤ)

It preserves the bilinear form, commutes with L, R, and sends {e_{i}, e_{i+1}} ⟼ {e_{i+1}, e_{i+2}} for all i ∈ ℤ. So, e_{i} = ϱe_{i−1} = ϱ^{i}e_{0} = z_{i}e_{0} − z_{i+1}e_{1} , where the integer numbers z_{i} form a sequence

…, (1 − h^{2}), −h − 1, 0, 1, h, (h^{2} − 1), …

such that z_{0} = 1, z_{1} = 0, hz_{i} = z_{i−1} + z_{i+1}. This sequence is skew symmetric w. r. t. i = 1, i. e. z_{i} = −z_{2−i}. Since

z_{i+1}^{2} − z_{i}z_{i+2} = det (\begin{matrix}z_{i} & z_{i+1} \\ −z_{i+1} & −z_{i+2} \end{matrix}) = det ϱ^{i} = 1

1in fact, Isom(M) is nothing but the group of integer units in the quadratic extension ℚ[\sqrt{h^{2} − 4}]

and

\frac{z_{i+1}}{z_{i+2}} − \frac{z_{i}}{z_{i+1}} = \frac{1}{z_{i+1}z_{i+2}} = \frac{1}{z_{i+1}^{2} − 1} ,

z_{i} are monotone increasing with lim_{i→±∞} z_{i} = ±∞ and {e_{i}, e_{i+1}} form a base for M ∀ i∈ℤ . At the same time, When i runs through ℤ from −∞ up to +∞, the point e_{i} = z_{i}e_{0} − z_{i+1}e_{1} runs consequently all integer points on the right upper branch of this hyperbola, because there are no integer points inside a parallelogram spanned by e_{i} and e_{i+1}. This gives all the statements of the proposition and also leads to the corollary from n^{∘} 1.7.2 below. □

1.7.2. COROLLARY. For any v ∈M with ⟨ v , v ⟩ > 0 there exists a unique i∈ℤ such that v = ae_{i} + be_{i+1} with integer a, b of the same sign and a ≠ 0. □

§2. Semiorthogonal decompositions in linear triangulated categories.

2.1. Tensoring and convolutions. A triangulated category � is called linear, if functor

Hom^{•}(X, Y) =^{def} ⊕_{ν} Hom^{ν}(X, Y)

takes values in the triangulated category of finite dimensional graded vector spaces. We will use the tensor product V^{•}⊗W^{•} and the dualization V^{•} ⟼ V^{×•} defined for graded vector spaces by the usual rules

Gr^{p}(V^{•}⊗W^{•}) =^{def} ⨁_{α+β=p} Gr^{α}(V^{•})⊗Gr^{β}(W^{•}) ; Gr^{p}(V^{×•}) =^{def} Gr^{−p}(V^{•})

If V^{•} is a graded vector space and X is an object of a linear triangulated category � , then the tensor product V^{•}⊗X is defined by the universal property

Hom^{•}(Y, V^{•}⊗X) = V^{•}⊗Hom^{•}(Y, X) ∀ Y ∈Ob�

and can be constructed as V^{•}⊗X = ⨁_{α} V^{α}⊗X[−α], where V^{α}⊗X[−α] is dim V^{α}-times di-

rect sum X[−α] with itself. A covariant exact (or cohomology) functor Cov^{•} between linear categories is called linear if Cov^{•}(V^{•}⊗X) = V^{•}⊗Cov^{•}(X). Similarly, a contravariant linear functor Cnt^{•} must satisfy the condition Cov(V^{•}⊗X) = V^{×•}⊗Cov(X). For example, Hom^{•}(∗, ∗) is bilinear, i. e.

Hom^{•}(W^{•}⊗X, V^{•}⊗Y) = W^{×•}⊗V^{•}⊗Hom^{•}(X, Y) .

This implies, in particular, that the identity endomorphism from

End(Hom^{•}(X, Y)) = Hom^{•}(Hom^{•}(X, Y)⊗X, Y) = Hom^{•}(X, Hom^{×•}(X, Y)⊗Y)

§ 2. Semiorthogonal decompositions in linear triangulated categories. 9

induces two canonical convolutions

\begin{aligned}j^{∗}(X, Y) & : & Hom^{•}(X, Y)⊗X ^{}Y \\ j_{∗}(X, Y) & : & X ^{}Hom^{×•}(X, Y)⊗Y . \end{aligned}  (2-1)

2.2. Exceptional objects and mutations. An object E ∈ Ob� is called exceptional, if Hom^{•}(E, E) is 1-dimensional algebra generated by the identity map. For an exceptional E and any X ∈� a left mutation L_{E}X and a right mutation R_{E}X of X w. r. t. E are defined by the distinguished triangles

L_{E}X ^{k^{∗}}^{}Hom^{•}(E, X) ⊗ E ^{j^{∗}}^{}X ^{i^{∗}}^{}L_{E}X[1]  (2-2)
R_{E}X[−1] ^{i_{∗}}^{}X ^{j_{∗}}^{}Hom^{× •}(X, E) ⊗ E ^{k_{∗}}^{}R_{E}X  (2-3)

extending the canonical maps j^{∗} = j^{∗}(E, X) and j_{∗} = j_{∗}(X, E) from (2-1). Applying Hom^{•}(E, ∗) to the triangle (2-2), we get the orthogonality relation

Hom^{•}(E, L_{E}X) = 0  (2-4)

since Hom^{•}(E, ∗) takes j^{∗}(E, X) to the identity map. Similarly, Hom^{•}(∗, E) applied the triangle (2-3) takes j_{∗}(E, X) to the identity map and gives the orthogonality relation

Hom^{•}(R_{E}X, E) = 0  (2-5)

Full triangulated subcategories

E^{⊥} =^{def} {X ∈ Ob� | Hom^{•}(E, X) = 0}
⊥E =^{def} {X ∈ Ob� | Hom^{•}(X, E) = 0}

are called right and left orthogonals to E in � . Taking E′ ∈ Ob^{⊥}E, E′′ ∈ Ob E^{⊥} and applying Hom^{•}(E′, ∗) and Hom^{•}(∗, E′′) to the both mutation triangles (2-2) and (2-3), we get for any X ∈ Ob� the following bifunctor isomorphisms

Hom^{•}(E′, X) = Hom^{•}(E′, R_{E}X)[−1] = Hom^{•}(E′, L_{E}X)[1] = Hom^{•}(L_{E}E′, L_{E}X) (2-6)
Hom^{•}(X, E′′) = Hom^{•}(L_{E}X, E′′)[−1] = Hom^{•}(R_{E}X, E′′)[1] = Hom^{•}(R_{E}X, R_{E}E′′) (2-7)

(the right identities follow from the left ones, because of L_{E}X ∈ Ob E^{⊥} and R_{E}XOb^{⊥}E). So, L_{E} and R_{E} are shifted ‘orthogonal projections’ onto the right and left orthogonals of E. More precisely, we have

2.2.1. PROPOSITION. � ^{X↦R_{E}X[−1]} ^{}^{⊥}E is the right adjoint functor^{1} to the inclusion ⊥E ^{⊂} ^{}� and � ^{X↦L_{E}X[1]} ^{}E^{⊥} is the left adjoint functor to the inclusion E^{⊥} ^{⊂} ^{}� . They satisfy the identities L_{E} ∘ R_{E} = L_{E}[1] and R_{E} ∘ L_{E} = R_{E}[−1] and their restrictions

L_{E}|_{^{⊥}E} : ^{⊥}E ^{}E^{⊥} and R_{E}|_{E^{⊥}} : E^{⊥} ^{}^{⊥}E

1Note that the third term of a distinguished triangle is not canonicaly defined, in general, by the other two terms. In our case L_{E}X and R_{E}X are unique up to unique isomorphism, because of the orthogonality conditions Hom^{•}(E, L_{E}X) = Hom^{•}(R_{E}X, E) = 0. Some generalizations can be founded in [BBD]

are inverse to each other isomorphisms between these subcategories. Proof. The functoriality of the both projectors L_{E}[1] and R_{E}[−1] follows form the relations Hom^{•}(E′, X) = Hom^{•}(E′, R_{E}X[−1]) and Hom^{•}(X, E′′) = Hom^{•}(L_{E}X[1], E′′), which are the parts of (2-6) and (2-7) and mean that the both functors are adjoint to the corresponding inclusions. Applying L_{E} and R_{E} to the mutation triangles (2-3) and (2-2), we get the identities L_{E} ∘ R_{E} = R_{E}[1] and R_{E} ∘ L_{E} = L_{E}[−1], because of R_{E}E = L_{E}E = 0. Further, (2-6) and (2-7) imply that both restrictions L_{E}|_{^{⊥}E} : ^{⊥}E ^{}E^{⊥} and R_{E}|_{E^{⊥}} : E^{⊥} ^{}^{⊥}E preserve Hom^{•}(∗, ∗)’s. Finally, R_{E}L_{E}E′′ = R_{E}E′′[−1] = E′′, if E′′ ∈ Ob E^{⊥}, and L_{E}R_{E}E′ = L_{E}E′[1] = E′, if E′ ∈ Ob^{⊥}E. □

2.2.2. PROPOSITION. Hom^{× •}(L_{E}X, E) = Hom^{•}(E, R_{E}X)[−1] functorially on X, E. Proof. Applying Hom^{•}(∗, E) to (2-2) and Hom^{•}(E, ∗) to (2-3), we get the following distinguished triangles of graded vector spaces

Hom^{•}(L_{E}X, E)[−1] ^{}Hom^{•}(X, E) ^{λ}^{}Hom^{× •}(E, X) ^{}Hom^{•}(L_{E}X, E)
Hom^{•}(E, R_{E}X) ^{}Hom^{× •}(X, E) ^{}^{ϱ} Hom^{•}(E, X) ^{}Hom^{•}(E, R_{E}X)[−1]

where the linear operators λ, ϱ satisfy for all φ ∈ Hom^{•}(X, E), ψ ∈ Hom^{•}(E, X) the identity

ϱψ(φ) · Id_{E} = φ ∘ ψ = λφ(ψ) · Id_{E} ,

because of the definition of the canonical convolutions (2-1). So, these triangles are mapped to each other by the ×-functor from n^{∘} 2.1. □

2.3. Mutations of exceptional pairs. An ordered pair {E_{1}, E_{2}} of exceptional objects is called exceptional or semiorthogonal, if Hom^{•}(E_{2}, E_{1}) = 0. Applying (2-6) and (2-7) to the category � generated^{1} by an exceptional pair {E_{1}, E_{2}}, we see that the pairs

L{E_{1}, E_{2}} =^{def} {L_{E_{1}}E_{2}, E_{1}} and R{E_{1}, E_{2}} =^{def} {E_{2}, R_{E_{2}}E_{1}}

are exceptional as well and generate the same category � . They are called the left mutation and right mutation of the initial pair {E_{1}, E_{2}}. Since R_{E_{1}}E_{2}[−1] = E_{2} and L_{E_{2}}E_{1}[1] = E_{1}, we have

Hom^{× •}(L_{E_{1}}E_{2}, E_{1}) = Hom^{•}(E_{1}, R_{E_{1}}E_{2}[−1]) = Hom^{•}(E_{1}, E_{2})
Hom^{•}(E_{2}, R_{E_{2}}E_{1}) = Hom^{× •}(L_{E_{2}}E_{1}[1], E_{1}) = Hom^{× •}(E_{1}, E_{2})

by (2-6), (2-7). Hence, the triangles

L_{E_{1}}E_{2}  ^{}Hom^{•}(E_{1}, E_{2}) ⊗ E_{1}  ^{}E_{2}  ^{}L_{E_{1}}E_{2}[1]
L_{E_{1}}E_{2}  ^{}Hom^{× •}(L_{E_{1}}E_{2}, E_{1}) ⊗ E_{1}  ^{}R_{E_{1}}L_{E_{1}}E_{2}  ^{}L_{E_{1}}E_{2}[1]

coincide canonically, that is R ∘ L {E_{1}, E_{2}} = {E_{1}, E_{2}}. Similarly, L ∘ R {E_{1}, E_{2}} = {E_{1}, E_{2}}. So, R and L act on the set of all exceptional pairs as inverse transformations.

1a category generated by a collection of objects E_{1}, E_{2}, …, E_{m} always means the smallest full triangulated subcategory containing all ‘linear combinations’ ⊕_{ν} Vν^{•} ⊗ E_{ν}, where Vν^{•} are graded vector spaces

2.4. Exceptional collections and braid group action. An ordered collection of exceptional objects {E_{1}, E_{2}, …, E_{k}} is called exceptional or semiorthogonal, if

Hom^{•}(E_{j}, E_{i}) = 0 ∀ i < j .

2.4.1. Mutations of an exceptional collection {E_{0}, E_{1}, …, E_{k}} are defined as follows.

For 1 ≤ i ≤ k a left mutation L_{i} replaces i-th pair of consequent elements {E_{i−1}, E_{i}} by its left mutation {L_{E_{i−1}}E_{i}, E_{i−1}}. A right mutation R_{i} replaces the same pair by its right mutation {E_{i}, R_{E_{i}}E_{i−1}}.

2.4.2. PROPOSITION. The mutations preserve the exceptionality and satisfy the relations

L_{i}R_{i} = R_{i}L_{i} = Id  (2-8)
L_{i}L_{j} = L_{j}L_{i} for |i − j| > 1  (2-9)
L_{i+1}L_{i}L_{i+1} = L_{i}L_{i+1}L_{i} for 1 < i < k  (2-10)

So, the braid group B_{k+1} acts by the mutations on exceptional collections of length (k + 1). Proof. Since the mutations L_{i} and R_{i} make changes only inside the category generated by the pair {E_{i−1}, E_{i}}, the exceptionality of the muted collection as well as the relation (2-8) follows from the discussion in the previous section. The relations (2-9) are evident. The

{A, B, C}

[figure: {L_{A}B, A, C}; L_{1}; {A, L_{B}C, B}; L_{2}]

[figure: {L_{A}B, L_{A}C, A}; L_{2}; {L_{A}L_{B}C, A, B}; L_{1}]

[figure: {L_{L_{A}B}L_{A}C, A, L_{A}B}; L_{2}; L_{1}]

Fig. A. The triangle relation (2-10) for the mutations

triangle equation (2-10) means that the diagram in fig. A on the page 11 is commutative for any exceptional triple {A, B, C}. The key point is to check the coincidence

L_{A}L_{B}C = L_{L_{A}B}L_{A}C

prescribed by the lower right arrow L_{2}. To do this we apply the exact linear functor L_{A} to the canonical triangle

L_{B}C ^{}Hom^{•}(B, C) ⊗ B ^{}C

and remember that Hom^{•}(B, C) = Hom^{•}(L_{A}B, L_{A}C) by (2-6). So, we have the exact triangle

L_{A}L_{B}C ^{}Hom^{•}(L_{A}B, L_{A}C) ⊗ L_{A}B ^{}L_{B}C

as required. □

2.5. Subcategories [E_{i} … E_{j}]. We write [E_{1} … E_{k}] for the full triangulated subcategory generated by an exceptional collection {E_{1}, E_{2}, …, E_{k}} in a linear triangulated category � . For any X ∈ Ob� consider compositions of its consequent mutations w. r. t. elements of the exceptional collection E_{1}, E_{2}, …, E_{k}

L_{[E_{1} ... E_{k}]} : X ⟼ L_{E_{1}}L_{E_{2}} … L_{E_{k}}X
R_{[E_{1} ... E_{k}]} : X ⟼ R_{E_{k}}R_{E_{k−1}} … R_{E_{1}}X

Applying inductively (2-6) and (2-7) to the consequent mutations, we get for all X ∈ Ob� , E′ ∈ Ob [E_{1} … E_{k}]^{⊥}, E′′ ∈ Ob [E_{1} … E_{k}]^{⊥} the functorial isomorphisms

Hom^{•}(E′, X) = Hom^{•}(E′, R_{[E_{1} ... E_{k}]}X)[−k] = Hom^{•}(E′, L_{[E_{1} ... E_{k}]}X)[k] = = Hom^{•}(L_{[E_{1} ... E_{k}]}E′, L_{[E_{1} ... E_{k}]}X) (2-11)

Hom^{•}(X, E′′) = Hom^{•}(L_{[E_{1} ... E_{k}]}X, E′′)[−k] = Hom^{•}(R_{[E_{1} ... E_{k}]}X, E′′)[k] = = Hom^{•}(R_{[E_{1} ... E_{k}]}X, R_{[E_{1} ... E_{k}]}E′′) (2-12)

The proposition from n^{∘} 2.2.1 is obviously extended as follows.

2.5.1. PROPOSITION. The functor � ^{X↦L_{[E_{1} ... E_{k}]}X[k]} ^{}[E_{1} … E_{k}]^{⊥} is left adjoint for

the inclusion [E_{1} … E_{k}]^{⊥} ^{⊂} ^{}� and the functor � ^{X↦R_{[E_{1} ... E_{k}]}X[−k]} ^{}[E_{1} … E_{k}]^{⊥} is

right adjoint for the inclusion [E_{1} … E_{k}]^{⊥} ^{⊂} ^{}� . They satisfy the relations

R_{[E_{1} ... E_{k}]}L_{[E_{1} ... E_{k}]} = R_{[E_{1} ... E_{k}]}[−k] and L_{[E_{1} ... E_{k}]}R_{[E_{1} ... E_{k}]}X = L_{[E_{1} ... E_{k}]}X[k] (2-13)

and their restrictions

L_{[E_{1} ... E_{k}]}| [E_{1} ... E_{k}]^{⊥} : [E_{1} … E_{k}]^{⊥}  ^{}[E_{1} … E_{k}]^{⊥}
R_{[E_{1} ... E_{k}]}|_{[E_{1} ... E_{k}]}_{⊥} : [E_{1} … E_{k}]^{⊥}  ^{}[E_{1} … E_{k}]^{⊥}

give inverse isomorphisms between the left and right orthogonals to [E_{1} … E_{k}]. Proof. Indeed, (2-11) implies that

Hom^{•}(E′, X) = Hom^{•}(E′, L_{[E_{1} ... E_{k}]}X)[k] = Hom^{•}(E′, R_{[E_{1} ... E_{k}]}L_{[E_{1} ... E_{k}]}X) ,

that is R_{[E_{1} ... E_{k}]}L_{[E_{1} ... E_{k}]} is the right adjoint functor to the inclusion [E_{1} … E_{k}]^{⊥} ^{⊂} ^{}� , i. e. it has to coincide with R_{[E_{1} ... E_{k}]}[−k]. The second relation in (2-13) is proved similarly. Other statements are proved using (2-11) (2-12) precisely like in n^{∘} 2.2.1. □

2.5.2. COROLLARY. The functors L_{[E_{1} ... E_{k}]} and R_{[E_{1} ... E_{k}]} depend only on the subcategory [E_{1} … E_{k}] but not on the particular choice of its semiorthonormal base E_{1}, E_{2}, …, E_{k} ). □

2.6. Dual exceptional bases. Let a linear triangulated category � be generated by an exceptional collection {E_{0}, E_{1}, …, E_{n}}. We will refer such a collection as an exceptional base of � . Two collections of objects {^{∨}E_{0},^{∨}E_{1}, …,^{∨}E_{n}} and {E_{0}^{∨}, E_{1}^{∨}, …, E_{n}^{∨}} are called respectively a left dual base and a right dual base if all Hom^{α}(^{∨}E_{β}, E_{γ}) and Hom^{α}(E_{β}, E_{γ}^{∨}) vanish except only for

Hom^{k} (^{∨}E_{k}, E_{n−k}) = Hom^{n−k} ( E_{n−k}, E_{k}^{∨}) = ℂ

Dual bases exist and can be constructed explicitly by consequent mutations as follows:

\begin{matrix}E_{k}^{∨} & =^{def} L_{[E_{0} ... E_{n−k−1}]}E_{n−k} = L_{E_{0}}L_{E_{1}} … L_{E_{n−k−1}}E_{n−k} \\ ^{∨}E_{k} & =^{def} R_{[E_{n−k+1} ... E_{n}]}E_{n−k} = R_{E_{n}}R_{E_{n−1}} … R_{E_{n−k+1}}E_{n−k} \end{matrix}  (2-14)

Indeed, L_{[E_{0} ... E_{n−k−1}]}E_{n−k} ∈ [E_{0} … E_{n−k−1}]^{⊥} and by (2-11) we also have

Hom^{•}(E_{ν}, L_{[E_{0} ... E_{n−k−1}]}E_{n−k}) = Hom^{•}(E_{ν}, E_{n−k})[−(n − k)] =

= \{ \begin{matrix}0 , if ν > (n − k) \\ ℂ in degree (n − k) for ν = (n − k) \end{matrix}

So, E_{k}^{∨} = L_{[E_{0} ... E_{n−k−1}]}E_{n−k} satisfies the orthogonality conditions. The arguments for ^{∨}E_{k} = R_{[E_{n−k+1} ... E_{n}]}E_{n−k} are quite similar. In fact, the dual bases are uniquely up to unique isomorphism defined by the above orthogonality conditions. This follows from

2.6.1. PROPOSITION. If {^{∨}E_{0},^{∨}E_{1}, …,^{∨}E_{n}} and {E_{0}^{∨}, E_{1}^{∨}, …, E_{n}^{∨}} satisfy the dual base orthogonality conditions, then ^{∨}E_{k} represents the covariant functor

X ⟼ Hom^{•}(E_{n−k}, L_{[E_{0} ... E_{n−k}]}X)

and E_{k}^{∨} represents the contravariant functor

X ⟼ Hom^{× •}(E_{n−k}, L_{[E_{0} ... E_{n−k}]}X)[−n] .

In particular, for any X ∈ Ob� there are functorial isomorphisms

Hom^{•}(^{∨}E_{k}, X)[n] = Hom^{× •}(X, E_{k}^{∨})  (2-15)

Proof. Since L_{[E_{0} ... E_{n}]}X ∈ [E_{0} … E_{n}]^{⊥} = 0, applying Hom^{•}(^{∨}E_{k}, ∗) and Hom^{•}(∗, E_{k}^{∨}) to the left mutation triangles

Hom^{•}(E_{ν}, L_{[E_{ν+1} ... E_{n}]}X) ⊗ E_{ν}  ^{}L_{[E_{ν+1} ... E_{n}]}X ^{}L_{[E_{ν} ... E_{n}]}X[1]  (2-16)

consequently for ν = 0, 1, …, (n − k − 1), we get

Hom^{•}(^{∨}E_{k}, L_{[E_{ν} ... E_{n}]}X) = Hom^{•}(L_{[E_{ν} ... E_{n}]}X, E_{k}^{∨}) = 0 for 0 ≤ ν ≤ (n − k) .

Hence, using (2-11) and applying Hom^{•}(^{∨}E_{k}, ∗) to (2-16) written for ν = n − k, we get

Hom^{•}(^{∨}E_{k}, X) = Hom^{•}(^{∨}E_{k}, L_{[E_{n−k+1} ... E_{n}]}X)[k] =

= Hom^{•}(E_{n−k}, L_{[E_{n−k+1} ... E_{n−k}]}X) ⊗ Hom^{•}(^{∨}E_{k}, E_{n−k}) =

= Hom^{•}(E_{n−k}, L_{[E_{n−k+1} ... E_{n−k}]}X)

because of Hom^{•}(^{∨}E_{k}, E_{n−k}) = ℂ[−k]. Similarly, applying Hom^{•}(∗, E_{k}^{∨}) to the same triangle and using (2-12), we get

Hom^{•}(X, E_{k}^{∨}) = Hom^{•}(L_{[E_{n−k+1} ... E_{n}]}X, E_{k}^{∨})[−k] =

= Hom^{× •}(E_{n−k}, L_{[E_{n−k+1} ... E_{n−k}]}X) ⊗ Hom^{•}(E_{n−k}, E_{k}^{∨})[−n] = = Hom^{× •}(E_{n−k}, L_{[E_{n−k+1} ... E_{n−k}]}X)[−n]

because of Hom^{•}(E_{n−k}, E_{k}^{∨}) = ℂ[−(n − k)]. □

2.6.2. COROLLARY. If � = [E_{0} … E_{n}] is generated by an exceptional collection, then there are the following isomorphisms of functors:

Hom^{•}(^{∨}E_{n−k}, X)[n] = = = = = Hom^{•}(E_{k}, L_{[E_{k+1} ... E_{n}]}X)[n]

Hom^{× •}(R_{[E_{0} ... E_{k−1}]}X, E_{k}) ‖  \begin{matrix}= \\ = \\ = \\ = \\ = \\ = \\ = \end{matrix} Hom^{× •}(X, E_{n−k}^{∨}) ‖

(2-17)

Proof. The upper and right identities was proved just before. If we change those arguments by their ‘right’ version, that is, apply Hom^{•}(^{∨}E_{k}, ∗) and Hom^{•}(∗, E_{k}^{∨}) to the triangle

R_{[E_{0} ... E_{k−1}]}X ^{}Hom^{× •}(R_{[E_{0} ... E_{k−1}]}X, E_{k}) ⊗ E_{k} ^{}R_{[E_{0} ... E_{k}]}X

and use (2-11), (2-12), and the identities

Hom^{•}(^{∨}E_{n−k}, R_{[E_{0} ... E_{k}]}X) = Hom^{•}(R_{[E_{0} ... E_{k}]}X, E_{n−k}^{∨}) = 0

which come by induction on [E_{ν} … E_{n}] starting from ν = 0, for which

R_{[E_{0} ... E_{n}]}X ∈ [E_{0} … E_{n}]^{⊥} = 0 ,

then we get

Hom^{× •}(R_{[E_{0} ... E_{k−1}]}X, E_{k}) =

= Hom^{× •}(R_{[E_{0} ... E_{k−1}]}X, E_{k}) ⊗ Hom^{•}(^{∨}E_{n−k}, E_{k})[n − k] =

= Hom^{•}(^{∨}E_{n−k}, R_{[E_{0} ... E_{k−1}]}X)[n − k] = Hom^{•}(^{∨}E_{k}, X)[n]

Hom^{•}(R_{[E_{0} ... E_{k−1}]}X, E_{k}) =

= Hom^{•}(R_{[E_{0} ... E_{k−1}]}X, E_{k}) ⊗ Hom^{•}(E_{k}, E_{n−k}^{∨})[k] =

= Hom^{•}(R_{[E_{0} ... E_{k−1}]}X, E_{n−k}^{∨})[k] = Hom^{•}(X, E_{n−k}^{∨})
□

2.7. Canonical Postnikov systems.. We continue to assume that � is generated by an exceptional collection {E_{0}, E_{1}, …, E_{n}} and write Vk^{•} for the graded vector spaces (2-17)

Vk^{•} =^{def} Hom^{•}(^{∨}E_{n−k}, X)[n] = Hom^{× •}(X, E_{n−k}^{∨}) =

= Hom^{•}(E_{k}, L_{[E_{k+1} ... E_{n}]}X)[n] = Hom^{× •}(R_{[E_{0} ... E_{k−1}]}X, E_{k}) ,

So, the triangles defining the consequent right mutations of X and the consequent left mutations of X[n] through [E_{0} … E_{n}] can be written as

\begin{matrix}V_{ν} ⊗ E_{ν}[−1] ^{k_{ν}}^{}R_{[E_{0} ... E_{ν}]}X[−1] ^{i_{ν}}^{}R_{[E_{0} ... E_{ν−1}]}X ^{j_{ν}}^{}V_{ν} ⊗ E_{ν} \\ V_{ν} ⊗ E_{ν} ^{j^{ν+1}}^{}L_{[E_{ν+1} ... E_{n}]}X[n] ^{i^{ν+1}}^{}L_{[E_{ν} ... E_{n}]}X[n + 1] ^{k^{ν+1}}^{}V_{ν} ⊗ E_{ν}[1] , \end{matrix}  (2-18)

where ν = 0, 1, …, n. We arrange them into a big diagram in fig. B on the page 15. At this diagram all oriented triangles along the left and right vertical borders are distinguished, the

§ 2. Semiorthogonal decompositions in linear triangulated categories. 15

morphisms i_{•} and i^{•} shown by broken arrows have degree 1, and all triangles and rhombuses in the central column are commutative. So, there is a complex (functorial in X)

0 ^{}V0^{•} ⊗ E_{0} ^{}V1^{•} ⊗ E_{1} ^{}⋯ ^{}Vn−1^{•} ⊗ E_{n−1} ^{}Vn^{•} ⊗ E_{n} ^{}0 (2-19)
R_{[E_{0} ... E_{n}]}X = 0  X[n]
[figure: V_{n} ⊗ E_{n}; j^{0}; k_{n}]
[figure: R_{[E_{0} ... E_{n−1}]}X; i_{n}; \begin{matrix}∼ \\ j_{n} \end{matrix}; L_{E_{n}}X[n]; i^{0}; k^{0}]
[figure: V_{n−1} ⊗ E_{n−1}; d_{n−1}; j^{1}; k_{n−1}]
[figure: R_{[E_{0} ... E_{n−2}]}X; i_{n−1}; j_{n−1}; L_{[E_{n−1} ... E_{n}]}X[n]; i^{1}; k^{1}]
[figure: V_{n−2} ⊗ E_{n−2}; d_{n−2}; j^{2}; k_{n−2}]
[figure: ⋮; i_{n−2}; ⋮; ⋮; i^{2}]
V_{2} ⊗ E_{2}
[figure: R_{[E_{0} ... E_{1}]}X; i_{2}; j_{2}; L_{[E_{2} ... E_{n}]}X[n]; i^{n−2}; k^{n−2}]
[figure: V_{1} ⊗ E_{1}; d_{1}; j^{n−1}; k_{1}]
[figure: R_{E_{0}}X; i_{1}; j_{1}; L_{[E_{1} ... E_{n}]}X[n]; i^{n−1}; k^{n−1}]
[figure: V_{0} ⊗ E_{0}; d_{0}; j^{n}; k_{0}]
[figure: X; i_{0}; j_{0}; 0 = L_{[E_{0} ... E_{n}]}X[n]; i^{n}; k^{n}]

Fig. B. Canonical Postnikov systems.

whose differentials Vp^{•} ⊗ E_{p} ^{d^{p}}^{}Vp+1^{•} ⊗ E_{p+1} equal d^{p} = k^{−p−1} ∘ j^{−p} = j_{n+p+2} ∘ k_{n+p+1}. The

left mutations produce a canonical left Postnikov system^{1}, which naturally identifies X with a canonical (left) convolution of this complex. Similarly, the right mutations produce a canonical (right) convolution^{2} of this complex and identify it with X as well. Let us outline again, that the both convolutions naturally depend on X.

2.7.1. Orthogonal decomposition. It follows from the diagram on fig. B on the page 15

that the orthogonal projector � ^{X↦L_{[E_{k+1} ... En]}X[n−k]} ^{}[E_{k+1} … E_{n}]^{⊥} from n^{∘} 2.5.1 takes its value inside the category [E_{0} … E_{k}], because the left mutation L_{[E_{k+1} ... E_{n}]}X is nothing but the canonical contraction of the complex (2-19) truncated on the right:

0 ^{}V0^{•} ⊗ E_{0}  ^{}V1^{•} ⊗ E_{1}  ^{}⋯ ^{}Vk−1^{•} ⊗ E_{k−1}  ^{}Vk^{•} ⊗ E_{k}  ^{}0

Since [E_{0} … E_{k}] ⊂ [E_{k+1} … E_{n}]^{⊥}, we have the coincidence

[E_{k+1} … E_{n}]^{⊥} = [E_{0} … E_{k}]  (2-20)

Similarly, the right mutation R_{[E_{0} ... E_{k}]}X equals the canonical contraction of the complex (2-19) truncated on the left:

0 ^{}Vk+1^{•} ⊗ E_{k+1}  ^{}Vk+2^{•} ⊗ E_{k+2}  ^{}⋯ ^{}Vn−1^{•} ⊗ E_{n−1}  ^{}Vn^{•} ⊗ E_{n}  ^{}0

So, the projector � ^{X↦R_{[E_{0} ... E_{k}]}X[−k−1]} ^{}[E_{0} … E_{k}]^{⊥} maps � into [E_{k+1} … E_{n}], that is

[E_{0} … E_{k}]^{⊥} = [E_{k+1} … E_{n}]  (2-21)

Moreover, for each k = 0, 1, …, n we have two natural transformations of functors:

R_{[E_{0} ... E_{k}]}X[−k − 1] ^{i_{k} ∘··· ∘ i_{0}} ^{}X ^{i^{n−k−1} ∘··· ∘ i^{0}} ^{}L_{[E_{k+1} ... E_{n}]}X[n − k]  (2-22)

2.7.2. COROLLARY. The triangle (2-22) is distinguished and gives a functorial orthogonal decomposition of any object X ∈ Ob� by the full triangulated subcategories [E_{0} … E_{k}]^{⊥} = [E_{k+1} … E_{n}] and [E_{k+1} … E_{n}]^{⊥} = [E_{0} … E_{k}].

1the Postnikov system notion was introduced in this context in [Kp], see also [GeMa, p. 257], although our indexing style here is slightly different — we proceed from the following observation: if we consider the first two terms of a distinguished triangle

E^{0} ^{d_{0}} E^{1} Con(d_{0})

in a homotopy category of complexes as a bicomplex with E^{0,0} situated at the origin, then the convolution of this complex coincides with the twisted cone Con(d_{0})[−1]; so, taking consequent cones from the left to the right in

[figure: E^{0}; E^{1}; E^{2}; ⋯; E^{n}; T^{0}; [1]; = = = = = = = = =; T^{1}[1]; [1]; ⋯; [1]; T^{n−1}[n − 1]; [1]; T^{n}[n]] we get the total complex T^{n} twisted by n

2or a right Postnikov system, see loc.cit.

Proof. Consider any Y ∈ Ob� introduced by distinguished triangle

Y [−1] ^{}R_{[E_{0} ... E_{k}]}X[−k − 1] ^{i_{k} ∘··· ∘ i_{0}} ^{}X ^{φ}^{}Y  (2-23)

For any Z′ ∈ Ob [E_{0} … E_{k}]^{⊥} = [E_{k+1} … E_{n}] we have the functorial isomorphism

Hom^{•}(Z, R_{[E_{0} ... E_{k}]}X[−k − 1]) =^{ψ↦i_{k} ∘··· ∘ i_{0} ∘ ψ} Hom^{•}(Z′, X)

given exactly by left multiplication by i_{k} ∘ ⋯ ∘ i_{0} (the functor X ↦ R_{[E_{0} ... E_{k}]}X[−k] is right adjoint to the inclusion [E_{0} … E_{k}]^{⊥} ^{⊂} ^{}� , see n^{∘} 2.5.1). Hence, Hom^{•}(Z′, Y) = 0 for any such Z′, i. e. Y ∈ Ob [E_{k+1} … E_{n}]^{⊥} = [E_{0} … E_{k}]. Now, taking Z′′ ∈ Ob [E_{k+1} … E_{n}]^{⊥} = [E_{0} … E_{k}] and applying Hom^{•}(∗, Z′′) to the triangle (2-23), we get an isomorphism

Hom^{•}(Y, Z′′) =^{ψ↦ψ ∘ φ} Hom^{•}(X, Z′′)

induced by right multiplication by φ. This means that the correspondence X ↦ Y is the left adjoint functor to the inclusion [E_{k+1} … E_{n}]^{⊥} ^{⊂} ^{}� , that is, the morphism X ^{φ}^{}Y coincides with the morphism X ^{i^{n−k−1} ∘··· ∘ i^{0}} ^{}L_{[E_{k+1} ... E_{n}]}X[n − k] (see n^{∘} 2.5.1). □

2.7.3. Spectral sequences. If we apply to the Postnikov system on fig. B on the page 15 a covariant cohomology functor Cov^{•} with values in some Abelian category, we get a bigraded coupled pair

D^{•,•}  \overline{^{Cov^{•}(i^{•})}} D^{•,•}

[figure: E^{•,•}; Cov^{•}(j^{•}); Cov^{•}(j^{•})]

with E^{p,q} = Cov^{q}(Vp^{•}⊗E_{p}) = ⨁_{α+β=q} V^{α} ⊗ Cov^{β}(E_{p}) and D^{p,q} = Cov^{q}(R_{[E_{0} ... E_{p−1}]}X). The mor-

phisms Cov^{•}(i_{·}), Cov^{•}(j_{·}), and Cov^{•}(k_{·}) have bidegrees (−1, 1), (0, 0), and (1, 0) respectively. The grading is bounded by 0 ≤ p, q ≤ n. So, there exists a spectral sequence (see [We, sec. 5.9]), which has E_{1}-term

E_{1}^{p,q} = ⨁_{α+β=q} V_{p}^{α} ⊗ Cov^{β}(E_{p})  (2-24)

where V_{α} = Hom^{n+α}(^{∨}E_{n−p}, X) = Hom^{−α}(X, E_{n−p}^{∨})^{∗}. Its limit terms E_{∞}^{n} = ⨁_{p+q=n} E_{∞}^{p,q} are

graded objects associated with some decreasing filtration

Cov^{n}(X) = F^{0}Cov^{n}(X) ⊃ F^{1}Cov^{n}(X) ⊃ F^{2}Cov^{n}(X) ⊃ ⋯ ⊃ F^{n}Cov^{n}(X) ⊃ 0 .

on Cov^{n}(X).

Dually, to compute the values of a contravariant cohomology functor Cnt^{p+q}(X) from� to an Abelian category, there is a spectral sequence situated in the strip −n ≤ p ≤ 0, 0 ≤ q ≤ n and having

E_{1}^{p,q} = ⨁_{α+β=q} V_{−p}^{×α} ⊗ Cnt^{β}(E_{−p}) .  (2-25)

where V_{−p}^{×α} = Hom^{α}(X, E_{n+p}^{∨}) = Hom^{−n−α}(^{∨}E_{n+p}, X)^{∗}. It converges to Cnt^{p+q}(X) in the same sense as above.

2.8. Helices. If� is generated by an exceptional collection {E_{0}, E_{1}, …, E_{n}}, then an infinite collection {E_{i}}_{i∈ℤ} defined by prescription

E_{i+n+1} = R_{[E_{i+1} ... E_{i+n}]}E_{i} and E_{i−n−1} = L_{[E_{i−n} ... E_{i−1}]}E_{i}

is called a helix of a period (n+1) with a foundation {E_{0}, E_{1}, …, E_{n}}. Each helix is uniquely recovered from any collection of (n + 1) its consequent elements {E_{i}, E_{i+1}, …, E_{i+n}} and all such collections are called the helix foundations as well. The braid group B_{n+1} acts on the helices of period (n+1) as follows. For i = 1, …, n mutations L_{i} and R_{i} replace all the pairs

{E_{i−1+k(n+1)}, E_{i+k(n+1)}} , where k ∈ℤ

by its left and right mutations. The same arguments as in n^{∘} 2.4 show that the muted helix is again a helix.

2.8.1. Serre duality. By (2-17), for any X and any i ∈ ℤ we have the canonical Serre duality isomorphism

Hom^{•}(E_{i}, X)[n] = Hom^{× •}(X, E_{i−n−1})  ∀ i∈ℤ .  (2-26)

Applying it twice, we also get the periodicity isomorphism

Hom^{•}(E_{i}, E_{j}) = Hom^{•}(E_{i−n−1}, E_{j−n−1}) ∀ i, j ∈ ℤ .  (2-27)

These isomorphisms mean that shifting an exceptional base by the helix period we get the Serre duality � ^{κ}^{}� . More precisely, consider two consequent helix foundations, say

E_{−n−1}, E_{−n}, …, E_{−2}, E_{−1} and E_{0}, E_{1}, …, E_{n−1}, E_{n}

Then a collection F_{0}, F_{1}, …, F_{n} with

F_{0} = E_{−1} = L_{[E_{0} ... E_{n−1}]}E_{n}
F_{n} = E_{0} = R_{[E_{−n} ... E_{−1}]}E_{−n−1}
F_{k} = R_{[E_{−k} ... E_{−1}]}E_{−k−1} = L_{[E_{0} ... E_{n−k−1}]}E_{n−k}

is simultaneously the left dual base for E_{−n−1}, E_{−n}, …, E_{−2}, E_{−1} and the right dual base for E_{0}, E_{1}, …, E_{n−1}, E_{n}. Now let us present an object X ∈ Ob� as a canonical convolution of the complex (2-19)

0 ^{}V0^{•} ⊗ E_{0}  ^{}V1^{•} ⊗ E_{1}  ^{}⋯ ^{}Vn−1^{•} ⊗ E_{n−1}  ^{}Vn^{•} ⊗ E_{n}  ^{}0

where Vk^{•} = Hom^{× •}(X, F_{n−k}). This complex together with the convolution procedure is completely defined by some tensor from the space

⊕_{p} Hom^{0} ( Vp^{•} ⊗ E_{p} , Vp+1^{•} ⊗ E_{p+1} ) = ⊕_{p,α} Hom^{−α}(Vp^{•}, Vp+1^{•}) ⊗ Hom^{α}(E_{p}, E_{p+1})

§ 3. Helix theory on ℙ_{n}. 19

Since Hom^{•}(E_{p}, E_{p+1}) = Hom^{•}(E_{p−n−1}, E_{p−n}), the same tensor produces a canonical convolution of the complex

0 ^{}V0^{•} ⊗ E_{−n−1}  ^{}V1^{•} ⊗ E_{−n}  ^{}⋯ ^{}Vn−1^{•} ⊗ E_{−2}  ^{}Vn^{•} ⊗ E_{−1}  ^{}0

We denote this convolution by κ(X).

2.8.2. PROPOSITION. Hom^{•}(Y, X) = Hom^{× •}(κ(X), Y)[n] as bifunctors on X, Y. Proof. Hom^{•}(Y, X) is computed by the functorial spectral sequence (2-24) with

E_{1}^{p,q} = ⨁_{α} V_{p}^{α} ⊗ Hom^{q−α}(Y, E_{p})

By (2-26), it is canonically dual to the spectral sequence (2-25)

E_{1}^{p,q} = ⨁_{α} V^{×α} ⊗ Hom^{n−q+α}(E_{−p−n−1}, Y)

which computes Hom^{•}(κ(X), Y)[n]. □

2.8.3. COROLLARY. Each helix {E_{i}}_{i∈ℤ} of period (n + 1) is periodic in the sence that E_{i−n−1} = κ(E_{n}), where � ^{κ}^{}� is the Serre dualizing functor defined above. □

§3. Helix theory on ℙ_{n}.

In this paragraph we always suppose � =� _{�oh}^{b} (ℙ_{n}) being a (bounded) derived category of coherent sheaves on the (complex) projective space. Historically, the helix theory was firstly developed in this category (see [Go1], [GoRu]).

3.1. Introductory example: invertible sheaves. The simplest helix on ℙ_{n} consists of invertible sheaves E_{n} =� (n), i∈ℤ; its period is (n + 1). To see this, consider the exceptional collection {� , � (1), …, , � (n)}. The consequent right mutations of � lay in the Abelian core � oh (ℙ_{n}) ⊂ � _{�oh}^{b} (ℙ_{n}), since the distinguished triangles (2-3) are reduced to the exact triples of sheaves. Indeed, the external powers of the Euler exact triple

[figure: 0; �; V ⊗� (1); �; 0]
[figure: 0; �; Λ^{2}V ⊗� (2); Λ^{2}�; 0]
⋮  ⋮  ⋮
[figure: 0; Λ^{k−1}�; Λ^{k}V ⊗� (k); Λ^{k}�; 0]
⋮  ⋮  ⋮
[figure: 0; Λ^{n−1}�; Λ^{n}V ⊗� (n); � (n + 1); 0]

where ℙ_{n} = ℙ(V), V = H^{0}(� (1))^{∗}, and � =� _{ℙ_{n}} is the tangent bundle, have

Λ^{k}V = Hom(� (k) , Λ^{k}� ) = Hom( Λ^{k−1}� , � (k) )^{∗}

and the maps coincide with the canonical convolutions (2-1). So,

R_{[� (1)...� (k)]}� = Λ^{k}� = L_{[� (k+1)...� (n)]}� (n + 1) .

and these relations are clearly ‘equivariant’ w. r. t. the twists. By the Serre theorem,�oh (ℙ_{n}) is spanned by invertible sheaves. Hence, {� , � (1), …, , � (n)} is an exceptional base for � _{�oh}^{b} (ℙ_{n}). In particular, the shift along the helix by the helix period should coincide with the dualizing functor (see n^{∘} 2.8.1), i. e. with the (anti)canonical twist. This agrees with the previous computation. Further, consider the following pair of dual bases for � _{�oh}^{b} (ℙ_{n})

{E_{0}, E_{1}, …, E_{n}} = { Λ^{n}Ω(n) , Λ^{n−1}Ω(n − 1), …, � } (with E_{p} = Λ^{n−p}Ω(n − p))
{^{∨}E_{0},^{∨}E_{1}, …,^{∨}E_{n}} = {� , � (1) , …, � (n) }  (with ^{∨}E_{p} =� (p))

and apply the functorial spectral sequence (2-24) to compute ‘the sheaves of cohomology’ functor

� ^{•} : � _{�oh}^{b} (ℙ_{n}) ^{}�oh (ℙ_{n})

which takes a complex of coherent sheaves to its sheaves of cohomologies. For any ‘pure’ object F ∈ Ob�oh (ℙ_{n}) we have

� ^{0}(F) = F and � ^{ν}(F) = 0 for ν ≠ 0 ,

and in (2-24) we obtain V_{i}^{α} = Ext^{n+α}(� (n − p), F) = H^{n+α}(F (p − n)). So, there is a spectral sequence with

E_{1}^{pq} = H^{n+q}(F (p − n)) ⊗ Λ^{n−p}Ω(n − p)

situated in the rectangle 0 ≤ p ≤ n, −n ≤ q ≤ 0 and converging to zero outside the main diagonal p + q = 0. The limit graded sheaf ⊕_{p=0}^{n} E_{∞}^{p,−p} at the main diagonal is associated

with some decreasing filtration on F. This is nothing but the well known Beilinson spectral sequence (maybe, in slightly unusual grading).

3.2. Strictness and purity. All known helices on ℙ_{n} have two extra features standing out the general setup of the previous paragraph. Namely, they are ‘pure’ and ‘strict’, that is, up to shifts through the derived category grading each helix consist of coherent (even locally free) sheaves satisfying Hom^{ν}(E_{i}, E_{j}) = 0 for all i < j and ν ≠ 0. Let us discuss these properties in more details.

3.2.1. Strict exceptional bases. An exceptional collection {E_{0}, E_{1}, …, E_{n}} ⊂� is called strict, if Hom^{q}(E_{i}, E_{j}) = 0 for all q ≠ 0 and i < j. A helix is called strict, if it has a strict foundation.

3.2.2. PROPOSITION. If a helix is strict, then all its foundations are strict and the strictness is preserved by the helix mutations. Proof. The first statement follows immediately from the periodicity (2-27) and Serre duality (2-26) isomorphisms:

Hom^{•}(E_{i}, E_{j}) = Hom^{•}(E_{i+n+1}, E_{j+n+1}) = Hom^{× •}(E_{j}, E_{i−n−1})[n]

To prove the second statement, we need some cohomology estimations. Let E be exceptional and X, Y — any objects of � . The long exact sequences for Hom^{•}(Y, ∗) and Hom^{•}(∗, Y) applied to the left and right mutation triangles:

X[−1] ^{}L_{E}X ^{}Hom^{•}(E, X) ⊗ E ^{}X
X ^{}Hom^{× •}(X, E) ⊗ E ^{}R_{E}X ^{}X[1]

give the following implications:

Hom^{>k}(E, Y) = Hom^{>k}(X, Y) = 0 ⟹ Hom^{>k}(L_{E}X, Y) = 0  (3-1)
Hom^{<k}(Y, E) = Hom^{<k}(Y, X) = 0 ⟹ Hom^{<k}(Y, L_{E}X) = 0  (3-2)
Hom^{>k}(Y, E) = Hom^{>k}(Y, X) = 0 ⟹ Hom^{>k}(Y, R_{E}X) = 0  (3-3)
Hom^{<k}(E, Y) = Hom^{<k}(X, Y) = 0 ⟹ Hom^{<k}(R_{E}X, Y) = 0  (3-4)

So, the smallest index ν for which Hom^{ν}(Y, X) ≠ 0 does not decrease under the left mutations of X and the right mutations of Y as well as the greatest index ν for which Hom^{ν}(Y, X) ≠ 0 does not increase under the right mutations of X and the left mutations of Y.

Now consider, say, the right mutation (E_{0}, E_{1}) ⟼ (E_{1}, R_{E_{1}}E_{0}). It is enough to show that all the exceptional pairs (E_{α}, R_{E_{1}}E_{0}) with −(n − 2) ≤ α ≤ −1 and α = 1 as well as all the exceptional pairs (R_{E_{1}}E_{0}, E_{β},) with 2 ≤ β ≤ n and β = n + 2 are strict. The above estimations show that Hom^{>0}(E_{α}, R_{E_{1}}E_{0}) = Hom^{>0}(R_{E_{1}}E_{0}, E_{β},) = 0. If we write

R_{E_{1}}E_{0} = L_{[E_{2} ... E_{n}]}E_{n+1} = L_{E_{2}}L_{E_{3}} ⋯ L_{E_{n}}E_{n+1}

then by (3-1) and (3-2) we get Hom^{<0}(E_{α}, R_{E_{1}}E_{0}) = Hom^{<0}(R_{E_{1}}E_{0}, E_{β}) = 0 as well. □

3.2.3. COROLLARY. Hom^{>0}(E_{i}, E_{j}) = Hom^{<n}(E_{j}, E_{i}) = 0 for all i < j in any strict helix {E_{i}}_{i∈ℤ}. Proof. By the Serre duality, it is enough to show that Hom^{>0}(E_{i}, E_{j}) = 0 for j ≥ i. This holds inside any foundation, i. e. for i ≤ j ≤ i + n; then we use an increasing induction on j −i: writing E_{j} = R_{[E_{j−n} ... E_{j−1}]}E_{j−n−1} and applying (3-3) to the consequent right mutations, we get inductively Hom^{>0}(E_{i}, E_{j}) = 0 for all j ≥ (i + n + 1). □

3.2.4. Pure exceptional objects. An object X ∈ Ob� _{�oh}^{b} (ℙ_{n}) is called pure, if it is isomorphic to F [k] for some coherent sheaf F and k ∈ℤ. Since an exceptional coherent sheaf has trivial unobstructed deformation theory with the one point moduli space, it has to be homogeneous. In particular, exceptional sheaves in � _{�oh}^{b} (ℙ_{n}) are locally free. There is also following very useful general criterium of purity established in [Po]:

3.2.5. PROPOSITION. Let E ∈ Ob� _{�oh}^{b} (X) be an object of the bounded derived category of coherent sheaves on a smooth algebraic variety X. If^{1} R� om(E, E) ∈ � ^{≤0}, then E is pure and locally free. Proof. It is the ‘main lemma’ of the paper [Po]. Its proof as well as some reformulations see there on the p.p. 5–6. □

1recall that � _{� oh}^{b} (X)^{opp} ×� _{� oh}^{b} (X) ^{R� om(∗,∗)} � _{� oh}^{b} (X) is the derived functor for the sheaves of local homomorphisms: if 0 P_{n} ⋯ P_{1} P_{0} F 0 is a locally free resolution for F ∈ Ob� oh (ℙ_{n}), then R�om(F, E) is represented by a complex of sheaves �om(P_{−p}, F)

3.2.6. COROLLARY. Any strict helix in � _{�oh}^{b} (ℙ_{n}) consist of locally free sheaves simultaneously subjected to the same grading translation. Moreover, non zero spaces Hom^{ν}(E_{i}, E_{j}) can appear only among Hom^{0}(E_{i}, E_{j}) with i > j and Hom^{n}(E_{i}, E_{j}) with j < i − n. Proof. We write H^{•} for the global cohomology spaces^{1}, � ^{•} ∈ Ob� oh (ℙ_{n}) — for the sheaves of cohomologies, and put� _{ij} =^{def} R�om(E_{i}, E_{j}). Let us compute Hom^{ν}(E_{i}, E_{j+k(n+1)}) for k ≫ 0. By the corollary from n^{∘} 3.2.3, these spaces vanish for all ν > 0. On the other hand, Hom^{ν}(E_{i}, E_{j}) = H^{ν}(� _{ij}) can be computed using the spectral sequence with E_{1}^{pq} = H^{p} (� ^{q}(� _{ij}) ) for the composition of derived functors (see [GeMa, p. 207, 219]). Since E_{j+k(n+1)} = E_{j} ⊗ ω^{∗}^{⊗k} and ω^{∗} is ample, this spectral sequence has E_{∞} = E_{1} for k large enough. In other words,

Hom^{ν}(E_{i}, E_{j+k(n+1)}) = H^{0} (� ^{ν}(� _{ij} ⊗ ω^{∗}^{⊗k}) ) .

Moreover, this space is non zero, if � _{ij} ≠ 0. Hence, � _{ij} ∈ � ^{≤0} ∀ i, j ∈ ℤ. Taking i = j and applying the proposition from n^{∘} 3.2.5, we see that E_{i} are shifted locally free sheaves. Since we also have � _{ij} = R� om(� _{ji},� ) ∈ � ^{≥0} as soon � _{ji} ∈ � ^{≤0}, we actually have � _{ij} ∈ � ^{0} = Ob� oh (ℙ_{n}) ∀ i, j ∈ ℤ. Hence, Hom^{<0}(E_{i}, E_{j}) = H^{<0}(� _{i,j}) = 0 and, by the Serre duality, Hom^{>n}(E_{i}, E_{j}) = 0 as well. Together with n^{∘} 3.2.3 this completes the proof. □

3.2.7. Open problems. We have seen that strict helices consist of locally free sheaves, have very nice cohomology properties, and remains to be strict after mutations. There are no examples of non strict helices on ℙ_{n} except for trivial ones obtained from the strict helices by shifting elements through the derived category grading. Also, there are no examples of exceptional objects different from (maybe shifted) locally free sheaves. These facts seems being of the same nature. The non-existence in the both cases could be deduced from the following (also open)

3.2.8. CONJECTURE. Any exceptional object in � _{�oh}^{b} (ℙ_{n}) can be included in a strict helix.

3.3. Strict helices. If we restrict ourselves by the strict helix area, then all the theory can be pulled down from � _{�oh}^{b} (ℙ_{n}) into �oh (ℙ_{n}). In what follows we always suppose that strict bases and/or helices are shifted to be consisting of pure sheaves situated in degree zero. Then for any i < j canonical distinguished triangles

R_{E_{j}}E_{i}[−1] ^{i_{•}}^{}E_{i} ^{j_{•}}^{}Hom^{× •}(E_{i}, E_{j}) ⊗ E_{j} ^{k_{•}}^{}R_{E_{j}}E_{i}
L_{E_{i}}E_{j} ^{k^{•}}^{}Hom^{•}(E_{i}, E_{j}) ⊗ E_{i} ^{j^{•}}^{}E_{j} ^{i^{•}}^{}L_{E_{i}}E_{j}[1]

turn to usual exact triples of locally free sheaves

\begin{matrix}0 ^{}E_{i}  ^{}Hom^{∗}(E_{i}, E_{j}) ⊗ E_{j}  ^{}R_{E_{j}}E_{i}  ^{}0 \\ 0 ^{}L_{E_{i}}E_{j}  ^{}Hom(E_{i}, E_{j}) ⊗ E_{i}  ^{}E_{j}  ^{}0 \end{matrix}  (3-5)

So, without loss of generality we could define the mutations inside a strict helix in terms of pure sheaves as it was done originally in [GoRu]. In particular, the left and right dual bases to any strict exceptional base consist of locally free sheaves as well.

1i. e. the derived functors for the global sections

3.3.1. Beilinson type spectral sequence. Let us compute a covariant cohomology functor

� ^{•} : � _{�oh}^{b} (ℙ_{n}) ^{}�oh (ℙ_{n})

which takes a complex X ∈ Ob� _{�oh}^{b} (ℙ_{n}) to its cohomology sheaves� (X) and acts identically on the pure shaves. The canonical spectral sequence (2-24) written for strictly exceptional base {E_{0}, E_{1}, …, E_{n}} and X = F ∈ Ob�oh (ℙ_{n}) may be arranged to have

E_{1}^{pq} = Ext^{q}(^{∨}E_{n+p}, F) ⊗ E_{n+p}  (3-6)

situated in the square −n ≤ p ≤ 0, 0 ≤ q ≤ n. It converges to a graded sheaf ⊕E_{∞}^{p,−p} situated on the main diagonal and associated with some decreasing filtration of F.

3.3.2. Eilenberg - Moore type spectral sequence. A contravariant functor Hom^{•}(∗, Y)

also can be computed via n^{∘} 2.7.3. Namely, let F_{1}, F_{2} be coherent sheaves and {E_{0}, E_{1}, …, E_{n}} be a strictly exceptional base. Then there is a spectral sequence with

E_{1}^{pq} = ⨁_{β−α=q} Ext^{α}(^{∨}E_{n−p}, F_{1})^{∗} ⊗ Ext^{β}(E_{n−p}, F_{2})  (3-7)

situated in the strip 0 ≤ p ≤ n, −n ≤ q ≤ n and converging to Ext^{p+q}(F_{1}, F_{2}).

3.3.3. Cohomology of a sheaf w. r. t. a helix. For a strict helix {E_{ν}} and any coherent

sheaf F on ℙ_{n} consider a table whose cells are indexed by the pairs of integers −∞ < p < +∞, 0 ≤ q ≤ n and the (p, q)-cell contains a vector space Ext^{q}(E_{p}, F). We call this table a left cohomology table of F w. r. t. {E_{ν}}. It is very important for cohomology calculations: for example, the both spectral sequences above are referred to this table. Since E_{i+k(n+1)} = E_{i} ⊗ ω^{∗}^{⊗k} and ω^{∗} is ample, we have

Ext^{>0}(E_{i}, F) = 0 for i ≪ 0  (3-8)
Ext^{<n}(E_{i}, F) = 0 for i ≫ 0  (3-9)

In other words, the cohomology table vanishes far on the right except for the upper row and vanishes far on the left except for the lower row. The placement of non-zero entries in the middle part of the table encodes an important information about F.

3.3.4. m-regularity w. r. t. a helix. A coherent sheaf F is called m-regular w. r. t. a helix {E_{i}}, if Ext^{ν}(E_{m+ν}, F) = 0 for ν = 1, 2, …, n. In other words, m-regularity means that the cohomology table contains an increasing sequence of consequent zeros

0 = Ext^{n}(E_{m+n}, F) 0

...

0 0

Hom(E_{m}, F) =?

For the simplest helix E_{i} = O(i) our notion of m-regularity turns (up to the sign of m) to the classical Mumford - Castelnuvo m-regularity. Main properties of Mumford - Castelnuvo m-regular sheaves remains to be true in our more general setup:

3.3.5. PROPOSITION. If a coherent sheaf F is m-regular w. r. t. a helix {E_{i}}, then it is k-regular for any k < m as well and the canonical map Hom(E_{m}, F) ⊗ E_{m} ^{can} ^{}F is surjective. Proof. The long exact sequence of Ext^{•}(∗, F) applied to a left mutation triangle

0 ^{}L_{E}X ^{}Hom(E, X) ⊗ E ^{}X ^{}0

gives the implication

Ext^{>(k+1)}(X, F) = Ext^{>k}(E, F) = 0 ⟹ Ext^{>k}(L_{E}X, F) = 0  (3-10)

If F is m-regular, then it follows from this implication that Ext^{>0}(L_{[E_{m+1} ... E_{m+k−1}]}E_{m+k}, F) = 0 for 2 ≤ k ≤ (n + 1). Hence, the Beilinson type spectral sequence (3-6)

E_{1}^{pq} = Ext^{q}(^{∨}E_{−p}, F) ⊗ E_{m+p} ,

which is written for the base {E_{m−n}, …, E_{m}} whose left dual base {^{∨}E_{0},^{∨}E_{1}, …,^{∨}E_{n}} consists of

∨E_{i} = R_{[E_{m−i+1} ... E_{m}]}E_{m−i} = L_{[E_{m+1} ... E_{m+n−i}]}E_{m+n−i+1} for 0 ≤ i ≤ n ,

turns to an exact resolvent

0 ^{}Hom(E_{m+1}, F) ⊗ E_{m−n}  ^{}Hom(^{∨}E_{n−1}, F) ⊗ E_{m−n+1}  ^{}⋯
⋯ ^{}Hom(^{∨}E_{1}, F) ⊗ E_{m−1}  ^{}Hom(E_{m}, F) ⊗ E_{m}  ^{can} ^{}F ^{}0

(we have substituted ^{∨}E_{n} = E_{m+1} and ^{∨}E_{0} = E_{m}). This gives the second assertion. Similarly, if we write E_{m+ν−k} = L_{[E_{m+ν−k+1} ... E_{m+ν−k+n}]}E_{m+ν−k+n+1} for k ≥ 1 and step by step apply implications (3-10) to the consequent left mutations, we get Ext^{ν}(E_{m+ν−k}, F) = 0 for all ν > 0 inductively on k. This gives the first assertion. □

§4. Helix theory on Del Pezzo surfaces.

4.1. Del Pezzo surfaces. We write S for a smooth regular complex projective surface S whose anticanonical class −K_{S} is ample. In other words, S is either the projective plane ℙ_{2}, or 2-dimensional quadric ℙ_{1} × ℙ_{1}, or a blow-up of ℙ_{2} at appropriate d ≤ 8 points. In the last case we will always write S_{d} instead of S. We fix a sequence of the blow-up’s

S_{d} ^{σ_{d}}^{}S_{d−1} ^{σ_{d−1}}^{}⋯ ^{σ_{2}}^{}S_{2} ^{σ_{1}}^{}ℙ^{2}

and write h and e_{1}, e_{2}, …, e_{d} for the proper preimage of the line on ℙ_{2} and the proper preimages of consequently blow-up’ed points (so, e_{d} is rational and irreducible). The grope Pic S_{d} is generated by h and e_{1}, …, e_{d} with h^{2} = 1, e_{i}^{2} = −1, he_{i} = e_{i}e_{j} = 0 ∀ i ≠ j. The canonical class of S_{d} equals K_{S_{d}} = −3h + e_{1} + ⋯ + e_{d}.

§ 4. Helix theory on Del Pezzo surfaces. 25

4.1.1. Riemann - Roch formula. A bilinear version of the Grothendieck - Hirtzebruch - Riemann - Roch formula on an algebraic surface X looks like

χ(E, F) = ∑_{ν} dim Ext^{ν}(E, F) =

= rk(E) rk(F) ( χ(�_{X}) + \frac{1}{2} (μ_{can}(F) − μ_{can}(E)) + q(F) + q(E) − \frac{c_{1}(E) · c_{1}(F)}{rk(E)rk(F)} ) , (4-1)

where μ_{can}(E) = \frac{−K_{X} · c_{1}(E)}{rk(E)} is called an (anticanonical) slope and q(E) = \frac{c_{1}^{2}(E) − 2 c_{2}(E)}{2 rk(E)}.

For our surfaces S the arithmetic genus χ(�_{S}) = 1. The constrains χ(E, E) = χ(F, F) = 1 lead to

q(E) = \frac{1}{2}(\frac{c_{1}^{2}(E) + 1}{rk^{2}(E)} − 1)

and simplify (4-1) for exceptional sheaves E, F to

χ(E, F) =

= \frac{rk(E) rk(F)}{2} ( μ_{can}(F) − μ_{can}(E) + \frac{1}{rk^{2}(E)} + \frac{1}{rk^{2}(F)} + ( \frac{c_{1}(F)}{rk(F)} − \frac{c_{1}(E)}{rk(E)} )^{2} ) . (4-2)

The helix theory on Del Pezzo surfaces has more complicated combinatorics than on ℙ_{n}. However, it also can be formulated in terms of pure sheaves and, moreover, the set of exceptional (even rigid) sheaves is completely described as well as the set of all helices.

We have to start with

4.1.2. THEOREM. The bounded derived category of coherent sheaves � _{�oh}^{b} (S) has an exceptional base, say the following:

{� , � (1) , � (2)} on ℙ_{2},

{� , � (1, 0) , � (0, 1) , � (1, 1)} on ℙ_{1} × ℙ_{1}, and

{�_{e_{1}}(−1), …, , �_{e_{d}}(−1), �_{S}, �_{S}(h), �_{S}(2h)} on S_{d}. Proof. Let X be a smooth projective variety such that� _{�oh}^{b} (X) admits an exceptional base {E_{1}, E_{2}, …, E_{k}}. There are two general results of D. O. Orlov (see [Or1]) about the derived categories� _{�oh}^{b} (ℙ(� ) ) and� _{�oh}^{b} (B_{Y} (X)), where ℙ(� ) ^{p}^{}X is a projective vector bundle over X and B_{Y} (X) ^{σ}^{}X is the blow up of X along a smooth submanifold Y ⊂ X. Namely, if � _{�oh}^{b} (X) admits an exceptional base {E_{1}, E_{2}, …, E_{k}}, then � _{�oh}^{b} (ℙ(� ) ) is generated by exceptional collection

{ E_{1}, …, E_{k} , E_{1}(1), …, E_{k}(1) , …, …, …, E_{1}(r), …, E_{k}(r) } ,

where r = rk(� )−1 and twisting is made by the tensor powers of relative�_{p}(1). In particular, if Y ⊂ X also admits an exceptional base for � _{�oh}^{b} (Y), then there exists an exceptional base in a derived category � _{�oh}^{b} (I), of sheaves on the exceptional divisor I = ℙ� _{Y/X}. The blow up diagram

[figure: I =ℙ� _{Y/X}; ⊂; i; B_{Y} (X); Y; π; ⊂; X; σ]

decomposes � _{�oh}^{b} (B_{Y} (X)) by two semiorthogonal full subcategories: Lσ^{∗} (� _{�oh}^{b} (X) ) and the right orthogonal to� inside Ri_{∗} (� _{�oh}^{b} (I) ). This allows to combine exceptional bases of � _{�oh}^{b} (I) and � _{�oh}^{b} (X) to get an exceptional base for � _{�oh}^{b} (B_{Y} (X)). We refer to [Or1] for all the details. □

4.2. Cohomological properties of sheaves on S. A coherent sheaf F on S is called rigid, if Ext^{1}(F, F) = 0. A rigid sheaf F is called extra rigid, if Ext^{2}(F, F) = 0 as well. A sheaf F is called simple, if Hom(E, E) = ℂ. So, ‘exceptional’ is ‘simple’ plus ‘extra rigid’. We write T F ⊂ F for the torsion subsheaf, write T^{0}F ⊂ T F for the torsion concentrated at points, and put T^{1}F =^{def} T F/T^{0}F (comp. with [HuLe, sec. 1.1]). Detailed proofs of all the results listed below may be found in [KuOr].

4.2.1. Mukai lemmas were established in [Mu]. They hold on any smooth projective surface and follow immediately from the spectral sequence^{1} with

E_{1}^{p,q} = ⊕_{β−α=p} Ext^{p+q}(Gr^{α}E, Gr^{β}E)  (4-3)

which computes Ext^{p+q}(E, E) using a decreasing filtration E = F^{0} ⊃ F^{1} ⋯ ⊃ F^{n+1} = 0 with quotients Gr^{i} = F^{i}/F^{i+1}.

4.2.2. LEMMA. Let 0 ^{}F ^{}E ^{}G ^{}0 be an exact triple of coherent sheaves on a smooth projective surface such that Hom(F, G) = Ext^{2}(G, F) = 0. Then

dim Ext^{1}(E, E) ≥ dim Ext^{1}(G, G) + dim Ext^{1}(F, F) .

Proof. Write (4-3) for 2-term filtration 0 ⊂ F ⊂ E with Gr^{0} = E/F and Gr^{1} = F. □

4.2.3. LEMMA. Let E be a torsion free coherent sheaf on a smooth projective surface. Then

dim Ext^{1}(E, E) ≥ dim Ext^{1}(E^{∗∗}, E^{∗∗}) + 2 length ( E^{∗∗}/E) .

Proof. Sinse E is torsion free and we deal with a surface, the reflexive span E^{∗∗}, of E, is locally free and contains E as a subsheaf. The factor E^{∗∗}/E haz zero dimensional support. It remains to write down (4-3) for 2-term filtration 0 ⊂ E ⊂ E^{∗∗} with Gr^{0} = E^{∗∗}/E and Gr^{1} = E. □

4.2.4. COROLLARY. Rigid torsion free sheaf is locally free on any projective surface.

4.2.5. Cohomology estimations. The next series of statements is specific for the Del Pezzo surfaces S listed in n^{∘} 4.1.

4.2.6. LEMMA. Let G, F be coherent sheaves on S. Then

dim Hom(F, G) ≥ dim Ext^{2}(G, F) .

Proof. Choose an anticanonical section φ ∈ H^{0}(� _{S}(−K_{S})) such that the corresponding anticanonical curve D =^{def} (φ)_{0} ∈ | − K_{S}| satisfies the conditions

dim(D ∩ Supp T^{1}G) < 1 and D ∩ Supp T^{0}G = ∅  (4-4)

1See [We] for details about all the spectral sequences we will use.

(such a curve exists, because | − K_{S}| has no base points and dim Supp T^{1}G = 1). If we apply �om(∗ , G) to the exact sequence

0 ^{}�_{S} ^{φ}^{}�_{S}(−K_{S}) ^{}�_{S}(−K_{S})|_{D} ^{}0

then we get an exact quadruple

0 ^{}�om(�_{S}(−K_{S})|_{D}, G) ^{}G(K_{S}) ^{}G ^{}�xt^{1}(�_{S}(−K_{S})|_{D}, G) ^{}0

because of�om(�_{S}(−K_{S}), G) ≅�_{S}(−K_{S})^{∗} ⊗ G = G(K_{S}). By the choice of D (see (4-4)), we have

�om(�_{S}(−K_{S})|_{D}, T^{1}G) =�om(�_{S}(−K_{S})|_{D}, T^{0}G) = 0 .

This implies � om(�_{S}(−K_{S})|_{D}, G) = � om(�_{S}(−K_{S})|_{D}, T G) = 0, i. e. the above quadruple turns to the exact triple

0 ^{}G(K_{S}) ^{}G ^{}�xt^{1}(�_{S}(−K_{S})|_{D}, G) ^{}0 ,

which gives an inclusion Hom(F, G(K_{S})) ⊂ Hom(F, G). Hence, by the Serre duality,

dim Ext^{2}(G, F) = dim Hom(F, G(K_{S}) ≤ dim Hom(F, G)

as required. □

4.2.7. LEMMA. Let F be a coherent sheaf on S such that Supp F ∩ C ≠ ∅ for some curve C ⊂ | − K_{S}| (this holds, for example, if rkF > 0). Then dim Hom(F, F) > dim Ext^{2}(F, F). Proof. Since�xt^{1}(� (−K_{S})|_{D}, G) ≅ G ⊗�_{D}, taking in the previous proof G = F, we get an exact triple

0 ^{}F (K_{S}) ^{}F ^{}F ⊗�_{D} ^{}0

If we choose D ∈ | − K_{S}| such that F ⊗�_{D} ≠ 0, then we get the non zero map

Hom(F, F) ^{}Hom(F, F ⊗�_{D}) ,

that is, the strict inclusion Hom(F, G(K_{S})) ⊊ Hom(F, G) , which implies the strict inequality dim Hom(F, F) > dim Ext^{2}(F, F). □

4.2.8. COROLLARY. Let G be a rigid sheaf on S. Then T^{0}G = 0 and both T G, G/T G are rigid as well. Proof. Take F = T E, G = E/T E in n^{∘} 4.2.2 and note that by n^{∘} 4.2.6

dim Ext^{2}(F E/T E, T E) ≥ dim Ext^{0}(T E, F E/T E) = 0

because there are no global Hom’s from a torsion sheaf to a torsion free one. □

4.3. Exceptional sheaves on S and exceptional objects in � _{� oh}^{b} (S). Combining cohomology estimations of the previous section with Riemann - Roch formula, we immediately obtain full geometric characterization of exceptional objects in � _{�oh}^{b} (S). Let us start with the exceptional torsion sheaves.

4.3.1. LEMMA. c_{1}^{2}(E) = −1 for any exceptional torsion sheaf E on S; moreover, E =�_{C}(d) for some irreducible rational curve C ⊂ S with C^{2} = −1. Proof. The Riemann - Roch formula (4-2) powered by the condition rkE = 0 gives

1 = χ(E, E) = rk^{2}(E) + (rk(E) − 1) c_{1}^{2}(E) − 2 rk(E) c_{2}(E) = −c_{1}^{2}(E).

Let C = Supp E. If C is irreducible, then E is locally free sheaf on C, because its torsion T^{0}E = 0 by n^{∘} 4.2.8. Write r for its rank. Then c_{1}(E) = rC and c_{1}(E)^{2} = r^{2}C^{2} = −1, i. e. r = 1. The genus formula 2g − 2 = C · (C + K_{S}) shows that 2g = C · K_{S} + 1 < 1, since −K_{S} is ample. So, C is rational curve with C^{2} = −1 and E =�_{C}(d) is a line bundle over C.

If C is reducible, consider one of its irreducible components C_{0} ⊂ C and restrict E from C onto C_{0}. We get an exact triple

0 ^{}E_{1}  ^{}E ^{}E_{0}  ^{}0,

where Supp E_{0} = C_{0}, Supp E_{1} = C_{1} = C \ C_{0} and both E_{0}, E_{1} have no torsion concentrated at points. So, Hom(E_{1}, E_{0}) = Ext^{2}(E_{0}, E_{1}) = 0 and by the Mukai lemmas we can write

dim Ext^{0}(E, E) = dim Ext^{0}(E_{0}, E_{0}) + dim Ext^{0}(E_{1}, E_{1}) + χ(E_{0}, E_{1})
dim Ext^{2}(E, E) = dim Ext^{2}(E_{0}, E_{0}) + dim Ext^{2}(E_{1}, E_{1}) + χ(E_{1}, E_{0}) .

Since E is exceptional, we also have

1 = χ(E, E) = dim Ext^{0}(E_{0}, E_{0}) − dim Ext^{2}(E_{0}, E_{0}) + χ(E_{0}, E_{1}) + + dim Ext^{0}(E_{1}, E_{1}) − dim Ext^{2}(E_{1}, E_{1}) − χ(E_{1}, E_{0}).

But the skew part of χ(E_{0}, E_{1}) has for exceptional torsion sheaves E_{0}, E_{1} an extremely simple Riemann - Roch expansion (see (4-2))

χ(E_{0}, E_{1}) − χ(E_{1}, E_{0}) = −K_{S} · (rk(E_{0}) c_{1}(E_{1}) − rk(E_{1}) c_{1}(E_{0})) = 0.

That is,

1 = ( dim Ext^{0}(E_{0}, E_{0}) − dim Ext^{2}(E_{0}, E_{0}) ) + ( dim Ext^{0}(E_{1}, E_{1}) − dim Ext^{2}(E_{1}, E_{1}) ) .

By n^{∘} 4.2.7, this is possible only if either C_{0} or C_{1} has the zero intersection with −K_{S}. But −K_{S} is ample, i.e., −K_{S}C_{0} > 0 and −K_{S}C_{1} > 0. Thus, C is irreducible. □

4.3.2. COROLLARY. An exceptional sheaf on S is either locally free or a torsion sheaf �_{C}(d), where C ⊂ S is irreducible rational curve with C^{2} = −1. □

4.3.3. THEOREM. All exceptional objects in � _{�oh}^{b} (S) are pure, i. e. are exhausted by exceptional coherent sheaves shifted through the derived category grading. Proof. Let E ∈ Ob� _{�oh}^{b} (S) be exceptional with the cohomology sheaves � ^{ν} = � ^{ν}(E). The Eilenberg - Moore spectral sequence has the form (see^{1} [GeMa, p. 263–264])

E_{1}^{pq} = ⊕_{α−β=p} Ext^{2p+q} (� ^{α},� ^{β} )

[figure: q; p; d; ∗; ∗; ∗; ∗; ∗; ∗; ∗; ∗; ∗; ∗; ∗; ∗; ∗; ∗; ∗; ∗; ∗; ∗; ∗; ∗; ∗]
Fig. C. Eilenberg - Moore’s E_{1}^{pq} = ⊕_{α−β=p} Ext^{2p+q} (� ^{α},� ^{β} )

and converges to Ext^{p+q}(E, E). All Ext^{1} (� ^{α},� ^{β} ) -terms are situated on the line 2p+q = 1 and fail at once in E_{∞} (see fig. C). In particular, E_{1}^{01} = ⊕Ext^{1} (� ^{ν},� ^{ν} ) = 0, i. e. all � ^{ν} are rigid sheaves. On the same time, the differential

d^{(−1,2)} : ⊕Hom (� ^{ν},� ^{ν+1} ) ^{}⊕ Ext^{2} (� ^{ν},� ^{ν} )

has to be an isomorphism and the differential

d^{(0,0)} : ⊕Hom (� ^{ν},� ^{ν} ) ^{}⊕ Ext^{2} (� ^{ν},� ^{ν−1} )

has to be an epimorphism with 1-dimensional kernel. But by n^{∘} 4.2.7 for each ν we have the inequality

dim Ext^{2} (� ^{ν},� ^{ν+1} ) ≤ dim Ext^{0} (� ^{ν+1},� ^{ν} )

and the strict inequality

dim Ext^{2} (� ^{ν},� ^{ν} ) < dim Ext^{0} (� ^{ν},� ^{ν} ) .

So, there is at most one ν with non zero � ^{ν}. □

1But be careful, because of indices misprints and two different indexing styles simultaneously used in this book during the spectral sequences discussion; we always follow the ‘cohomology’ indexation, where the differential in E_{1}^{p,q} has bidegree (1, 0) and goes right down in the consequent terms E_{2}, E_{3}, …; it requires decreasing filtrations ⋯ ⊃ F^{p} ⊃ F^{p+1} ⊃ ⋯, for example, the Eilenberg - Moore spectral sequence comes from the canonical filtration F_{can}^{p}, which has Gr_{can}^{p}X =� ^{−p}(X)[p] and filters Hom(X, Y) with

Gr^{p}Hom(X, Y) = ⊕_{α−β=p} Hom(Gr^{α}X, Gr^{β}Y)[p] ,

so, taking H^{p+q} we get Ext^{2p+q}; the good textbook on spectral sequences and coupled pairs machinery is [We]

4.4. Mutations of exceptional sheaves. Let (E_{1}, E_{2}) be an exceptional pair of sheaves on S. Since all exceptional objects in � _{�oh}^{b} (S) are pure, the mutations L_{E_{1}}E_{2}, R_{E_{2}}E_{1} are pure sheaves and the triangles (2-2), (2-3) turns either into exact triples^{1}

\begin{matrix}0 ^{}L_{E_{1}}E_{2}  ^{}Hom(E_{1}, E_{2}) ⊗ E_{1}  ^{}E_{2}  ^{}0, \\ 0 ^{}E_{1}  ^{}Hom(E_{1}, E_{2})^{∗} ⊗ E_{2}  ^{}R_{E_{2}}E_{1}  ^{}0, \end{matrix}  (4-5)

or into more complicated sequences

0 ^{}Hom(E_{1}, E_{2}) ⊗ E_{1}  ^{}E_{2}  ^{}L_{E_{1}}E_{2}[1] ^{}Ext^{1}(E_{1}, E_{2}) ⊗ E_{1}  ^{}0,
0 ^{}Ext^{1}(E_{1}, E_{2})^{∗} ⊗ E_{1}  ^{}R_{E_{2}}E_{1}[−1] ^{}E_{1}  ^{}Hom(E_{1}, E_{2})^{∗} ⊗ E_{2}  ^{}0.

(4-6)

In particular, we automatically have Ext^{2}(E_{1}, E_{2}) = 0. The mutations (4-5) are called regular. The mutations (4-6) are called irregular. We will prove later (see the proposition from n^{∘} 5.3.5) that at least one of the spaces Hom(E_{1}, E_{2}), Ext^{1}(E_{1}, E_{2}) should vanish for any exceptional pair on Del Pezzo surface, i. e. the quadruples (4-6) turn actually to exact triples

0 ^{}Hom(E_{1}, E_{2}) ⊗ E_{1}  ^{}E_{2}  ^{}L_{E_{1}}E_{2}[1] ^{}0,
0 ^{}E_{2}  ^{}L_{E_{1}}E_{2}[1] ^{}Ext^{1}(E_{1}, E_{2}) ⊗ E_{1}  ^{}0,
0 ^{}R_{E_{2}}E_{1}[−1] ^{}E_{1}  ^{}Hom(E_{1}, E_{2})^{∗} ⊗ E_{2}  ^{}0,
0 ^{}Ext^{1}(E_{1}, E_{2})^{∗} ⊗ E_{1}  ^{}R_{E_{2}}E_{1}[−1] ^{}E_{1}  ^{}0.

(4-7)

4.4.1. COROLLARY. If Hom(E_{1}, E_{2}) ≠ 0, then each canonical contractions

Hom(E_{1}, E_{2}) ⊗ E_{1}  ^{}E_{2} E_{1}  ^{}Hom(E_{1}, E_{2})^{∗} ⊗ E_{2}

is either monomorphic or epimorphic.

4.5. Dual exceptional bases. Let an exceptional collection of sheaves {E_{0}, E_{1}, …, E_{k}} be a base for� _{�oh}^{b} (S). Then its right dual base {E_{0}^{∨}, E_{1}^{∨}, …, E_{k}^{∨}} consists of shifted pure sheaves, i. e. we can write

E_{i}^{∨} = L_{E_{0}}L_{E_{1}} ⋯ L_{E_{k−i−1}}E_{k−i} = E_{i}^{×}[Δ_{i}^{×}],

where E_{i}^{×} is an exceptional sheaf and Δ_{i}^{×} equals a number of irregular left mutations (4-7) in this series. Symmetrically, we write

∨E_{i} = R_{E_{k−i+1}}L_{E_{k−i+2}} ⋯ L_{E_{k}}E_{k−i} = ^{×}E_{i}[−^{×}Δ_{i}]

where ^{×}E_{i} is an exceptional sheaf and ^{×}Δ_{i} is number of irregular right mutations in this series (certainly, ^{×}Δ_{i} may differ from Δ_{i}^{×}). We have

Ext^{i−^{×}Δ_{i}}(^{×}E_{i}, E_{k−i}) = Ext^{k−i−^{×}Δ_{i}}(E_{k−i}, E_{i}^{×}) = ℂ

and all other Ext^{ν}(^{∨}E_{α}, E_{β}) = Ext^{ν}(E_{α}, E_{β}^{∨}) = 0.

1as it was on ℙ_{n} considered in the previous paragraph

4.5.1. Beilinson-type spectral sequence written in terms of pure sheaves is situated in the strip −k ≤ p ≤ 0, 0 ≤ q ≤ k and has

E_{1}^{p,q} = Ext^{q−^{×}Δ_{p}}(E_{−p}, F) ⊗^{×}E_{k+p} ,

if the left dual base is used, or

E_{1}^{p,q} = Ext^{k−q−Δ_{p}^{×}}(F, E_{−p})^{∗} ⊗ E_{k+p}^{×}

if you prefer the right one. The both depend functorially on a coherent sheaf F on S and converges to F in the usual sense: F admits a decreasing filtration with consequent quotients

E_{∞}^{0,0}, E_{∞}^{−1,1}, …, E_{∞}^{−k,k}

and E_{∞}^{p,q} = 0 for p + q ≠ 0 at the same time.

4.5.2. Example. Consider the simplest helix spanned by the foundation

{E_{0}, E_{1}, …, E_{3}} = {� , � (1, 0), � (0, 1), � (1, 1)}

on the quadric surface S = ℙ_{1} × ℙ_{1} (we write � (a, b) for p_{1}^{∗}�_{ℙ_{1}}(a) ⊗_{�_{S}}p_{2}^{∗}�_{ℙ_{1}}(b), where

ℙ_{1} ^{}^{p_{1}} ℙ_{1} × ℙ_{1} ^{p_{2}}^{}ℙ_{1}

are the standard projections). Then the right dual base has, certainly, E_{0}^{×} = E_{3} =� (1, 1) and E_{3}^{×} = E_{3} ⊗ ω_{S} = � (−1, −1). Further, E_{1}^{×} = � (−1, 0) because the corresponding mutation comes essentially from ℙ_{1} = ℙ(V):

0 ^{}� (−1, 0) ^{}V^{∗} ⊗� ^{}� (1, 0) ^{}0

(in particular, it is regular, that is Δ_{1}^{×} = 0). Finally, E_{2}^{×} = � (0, −1) with Δ_{2}^{×} = 1. Indeed, since the pair {E_{1}, E_{2}} = {� (1, 0), � (0, 1)} is completely orthogonal, the first mutation L_{E_{1}}E_{2} = E_{2} is the trivial irregular one; the next mutation (through E_{0}) comes again from ℙ_{1}

0 ^{}� (0, −1) ^{}V^{∗} ⊗� ^{}� (0, 1) ^{}0

and is regular. Note that Δ_{3}^{×} = 1, because the straightforward computation of E_{3}^{×} requires the consequent mutations

0 ^{}� (−1, 1) ^{}V^{∗} ⊗� (0, 1) ^{}� (1, 1) ^{}0
0 ^{}� (−1, 1) ^{}� _{S}(−1, −1) ^{}Λ^{2}V^{∗} ⊗� (1, 0) ^{}0
0 ^{}� (−1, −1) ^{}S^{2}V^{∗} ⊗� ^{}� _{S}(−1, −1) ^{}0

where the first and the third of are regular but the second is irregular and coincides with the canonical extension using

Ext^{1}(E_{1}, L_{E_{2}}E_{3}) = Ext^{1}(� (1, 0), O(−1, 1)) = Λ^{2}V^{∗} = ℂ .

So, the Beilinson spectral sequence presents an arbitrary coherent sheaf F on ℙ_{1} × ℙ_{1} by the functorial spectral sequence whose E_{1} term is situated in the rectangle −3 ≤ p ≤ 0, 0 ≤ q ≤ 3 and looks as

\begin{matrix}H^{2}(F (1, 1)) ⊗� (−1, −1) & H^{2}(F (0, 1)) ⊗� (0, −1) & 0 & 0 \\ H^{1}(F (1, 1)) ⊗� (−1, −1) & H^{1}(F (0, 1)) ⊗� (0, −1) & H^{2}(F (1, 0)) ⊗� (−1, 0) & H^{2}(F) ⊗� \\ H^{0}(F (1, 1)) ⊗� (−1, −1) & H^{0}(F (0, 1)) ⊗� (0, −1) & H^{1}(F (1, 0)) ⊗� (−1, 0) & H^{1}(F) ⊗� \\ 0 & 0 & H^{0}(F (1, 0)) ⊗� (−1, 0) & H^{0}(F) ⊗� \end{matrix}

with zeros at all other cells.

4.6. Transitivity of the braid group action. The main advance of the Helix theory on Del Pezzo surfaces is given by the following theorem. Recall that an exceptional collection is called full, if it generates � _{�oh}^{b} (S) as a triangulated category.

4.6.1. THEOREM. Any exceptional collection of sheaves on S may be extended to a full collection. The braid group acts transitively on the set of full exceptional collections of sheaves.

In particular, this theorem implies that any exceptional sheaf on Del Pezzo surface may be constructed by the mutations starting from some standard initial collection, say one of listed in n^{∘} 4.1. The proof of this theorem was published in [Ku3] and [KuOr]. It is long enough: besides a number of pure combinatorial geometrical tricks, similar to the ones used in the integer linear programming, it uses essentially an interlaying between several stability concepts, which will be discussed briefly in the next paragraph. Then, in §6, we will discuss the proof of the theorem n^{∘} 4.6.1 itself.

§5. Del Pezzo surfaces: stability of rigid sheaves.

5.1. Slopes. Let �_{tf} ( X) be a set of all torsion free coherent sheaves on a smooth algebraic variety X. It is partially ordered by inclusions. A function τ : �_{tf} ( X) ^{}Γ with values in a partially ordered set Γ is called a slope, if it has the following properties:

swing quality: for any exact triple of torsion free sheaves

0 ^{}F ^{}E ^{}G ^{}0

there is a triple of equivalences:

τ(F) = τ(E) ⟺ τ(E) = τ(G)
τ(F) ≤ τ(E) ⟺ τ(E) ≤ τ(G)
τ(F) ≥ τ(E) ⟺ τ(E) ≥ τ(G)

local finiteness: for any r_{0} ∈ ℕ and τ_{0} ∈ Γ there are no infinite sequences of torsion free sheaves G_{1}, G_{2}, G_{3}, … with bounded ranks: rkG_{i} ≤ r_{0} and monotone decreasing

slopes τ(G_{i}) > τ(G_{i+1}) bounded below: τ(G^{i}) > τ_{0} ;

§ 5. Del Pezzo surfaces: stability of rigid sheaves. 33

week monotony: τ(F) ≤ τ(E) for all F ⊂ G of the same rkF = rkG.

5.1.1. The most common way for producing the slopes is to use an additive function of

a coherent sheave divided by the rank. If Γ is an ordered Abelian group and α : K_{0}(X) ^{}Γ is an additive function, then τ(E) =^{def} α(E)/rk(E) is locally finite, because

|τ(E) − τ(F)| ≥ \frac{1}{max(rkE, rkF)^{2}} ,

and has the swing quality, since a sign of τ(E) − τ(F) equals a sign of the determinant

det (\begin{matrix}α(E) & α(F) \\ rk(E) & rk(F) \end{matrix}) = − det (\begin{matrix}(α(E) − α(F)) & α(E) \\ (rk(E) − rk(F)) & rk(E) \end{matrix}) .

The week monotony requires a more accurate choice of the additive function α.

5.1.2. Giesecker type slopes. For any ample divisor A and any coherent sheave E on X a Hilbert polynomial h_{E,A} ∈ ℚ[t] is defined by the property

h_{E,A}(n) =^{def} χ(E(nA)) = ∑_{ν=0}^{dim X} dim H^{ν}(E ⊗� (nA)) ∀ n∈ℕ .

We order ℚ[t] by prescription f ≤ g ⟺ f(n) ≤ g(n) ∀ n ≫ 0 and define a Giesecker slope of a torsion free sheaf E as

γ_{A}(E) =^{def} \frac{h_{E,A}}{r(E)} ∈ ℚ[t].

The Giesecker slope satisfies the strict version of the week monotony: if F ⊊ E are torsion free of the same rank, then γ_{A}(F) < γ_{A}(E), since χ((E/F)(nA)) > 0 for E/F ≠ 0 by the Serre theorem. Any slope that satisfies this strict version of the week monotony is called strict (or Giesecker type) slope.

5.1.3. Mumford - Takemoto slope. For any ‘nef’ divisor H on X define a Mumford - Takemoto slope of a torsion free sheaf E as

μ_{H}(E) =^{def} \frac{c_{1}(E) · H^{dim X−1}}{r(E)} .

This slope also satisfies little more precise property than the week monotony. Namely, if F ⊊ E are torsion free of the same rank, then μ_{H}(F) ≤ μ_{H}(E) with the equality possible only if:

codim supp (E/F) ≥ 1 (for ‘nef’ H) ,  (5-1)
codim supp (E/F) ≥ 2 (for ample H) .  (5-2)

This follows from the ‘nef’ property^{1} and from the Nakai - Moishezon criterium, because of c_{1}(E/F) is either zero or effective.

1By the definition, this means that D · H^{dim X−1} ≥ 0 for any effective divisor D.

5.2. Stability. Let τ be a slope. A torsion free sheaf E on X is called τ-stable (corr. τ-se-mistable), if for any subsheaf F ⊂ E with rkF < rkE the inequality

τ(F) < τ(E) (corr. τ(F) ≤ τ(E))

holds. By the swing quality, this inequalities are equivalent to the opposite inequalities

τ(G) > τ(E) (corr. τ(G) ≥ τ(E))

satisfied for all torsion free quotients G = E/F. Indeed, the standard ‘torsion removing’ diagram

[figure: T; F′; ⊂; E; G′; F; ∪; ⊂; E; ‖; G; T; ∪],

where T = Tors G, shows that it is enough to test (semi) stability only by subsheaves F′ ⊂ E with torsion free quotients G′ = E/F′.

5.2.1. Properties of semistable sheaves. Let F ^{φ} E be a morphism between τ-

semistable sheaves. The swing quality applied to the triple ker φ^{⊂} ^{}F [figure] im φ, taking into account the inclusion im φ ⊂ E, leads to the following implications:

τ(F) > τ(E) ⟹ φ = 0;
τ(F) = τ(E) and F is stable ⟹ φ is either zero or injective;
τ(F) = τ(E) and E is stable ⟹ φ is either zero or surjective at generic point,

Moreover, if γ is a Giesecker type slope, then the third implication becomes stronger

γ(F) = γ(E) and E is stable ⟹ φ is either zero or an epimorphism.

In particular, each stable sheaf E is simple, i. e. Hom(E, E) = ℂ.

Certainly, all invertible sheaves are stable just by the definition. Of course, semistablity is preserved under twists by invertible sheaves and tensoring by vector spaces. More generally, if an exact triple

0 ^{}F ^{}E ^{}G ^{}0

consists of torsion free sheaves of the same slope, then E is semistable iff both F, G are. A complete proof of these facts can be found in [Ku3]. Since any torsion free sheaf E on a projective variety is a quotient sheaf of a semistable

sheaf Hom(� (nA), E) ⊗� (nA), where A is an ample divisor and n is large enough, the slopes τ(G) of torsion free quotients G = E/F are bounded below, say by τ(� (nA)).

5.2.2. Harder - Narasimhan filtration. Any torsion free sheaf E has a canonical^{1} decreasing filtration

E = F^{0} ⊃ F^{1} ⊃ ⋯ ⊃ F^{n} ⊃ F^{n+1} = 0

with semistable quotients G^{i} = F^{i}/F^{i+1} of strictly increasing slopes τ(G^{i}) < τ(G^{i+1}). It is called the Harder - Narasimhan filtration.

It is constructed inductively as follows. Since the slopes of torsion free quotients G = E/F are bounded below, the local finiteness property implies that there exists minimal such the slope τ_{0}. Let G^{1} = E/F^{1} be a torsion free sheaf of the maximal rank among all torsion free quotients E/F with τ(E/F) = τ_{0}. Then G^{1} is semistable, has τ(G^{1}) > τ(E) > τ(F^{1}), and we can repeat the procedure with F^{1} instead of E as soon F^{1} is non semistable.

The uniqueness is established by a little more delicate induction going in the opposite direction. Namely, if (G′_{0}, G′_{1}, G′_{2}, …, G′_{m}) and (G′′_{0}, G′′_{1}, G′′_{2}, …, G′′_{n}) both satisfy the Harder - Narasimhan conditions, then G′_{m} ⊂ E and G′′_{n} ⊂ E are the subsheaves of the same slope τ(G′_{m}) = τ(G′′_{n}). Indeed, an inequality, say τ(G′_{m}) > τ(G′′_{n}), implies that Hom(G′_{m}, G′′_{i}) = 0 for all i by the semistability, i. e. Hom(G′_{m}, E) = 0. Further, an easy increasing induction on i = 0, 1, … shows that G′_{m} ⊂ F′′_{i} for all 0 ≤ i ≤ n, because Hom(G′_{m}, G′′_{i}) = 0 implies that G′_{m} ⊂ F′′_{i} is (mod F′′_{i+1})-congruent to zero. Similarly, G′′_{n} ⊂ F′_{j} for all 0 ≤ j ≤ m. Taking i = n and j = m, we get G′_{m} = G′′_{n} and can repeat the procedure for E/Q filtered with the quotients (G′_{0}/Q , …, G′_{m−1}/Q) and (G′′_{0}/Q , …, G′′_{n−1}/Q), where Q = G′_{m} = G′′_{n}. We refer the readers to [HuLe] for details.

5.2.3. Jordan - Holder filtration. We need its rather more precise version than the usual one (comp. with [HuLe]). Let γ be a Giesecker type slope, i. e. satisfy the strict inequality γ(F) < γ(E) for all torsion free F ⊊ E of the same rank. Then any γ-semistable torsion free sheaf E admits a decreasing filtration

E = F^{0} ⊃ F^{1} ⊃ F^{2} ⊃ ⋯ ⊃ F^{n} ⊃ F^{n+1} = 0  (5-3)

whose quotients G^{i} = F^{i}/F^{i+1} are semistable, have γ(G^{i}) = γ(F^{i}) = γ(E), satisfy the orthogonality conditions:

Hom ( F^{i+1}, G^{i} ) = Hom ( G^{i}, G^{i+1} ) = Hom ( G^{i+1}, G^{i} ) = 0  (5-4)

and each G^{i} for one’s turn admits a filtration

G^{i} = F^{i,0} ⊃ F^{i,1} ⊃ F^{i,2} ⊃ ⋯ ⊃ F^{i,m_{i}} ⊃ F^{i,m_{i}+1} = 0  (5-5)

with γ-stable isotype quotients Q^{i} = F^{i,ν}/F^{i,ν+1} (the same for all ν) with γ(Q^{i}) = γ(E). We will call the filtration (5-3) an isotype Jordan - Holder filtration.

To construct it, consider a torsion free quotient Q^{0}, of E, with γ(Q^{0}) = γ(E) and minimal possible rank. Obviously, it is stable. Write F̃^{1,0} for a kernel of the epimorphism E [figure] Q^{0}. It is semistable with γ(F̃^{1,0}) = γ(E). If Hom(F̃^{1,0}, Q^{0}) = 0, we put F^{1} = F̃^{1,0}. Otherwise, there is an epimorphism F̃^{1,0} [figure] Q^{0}. We write F̃^{1,1} for its kernel, which is semistable with γ(F̃^{1,1}) = γ(E). Repeating this procedure, we get a filtration

E = F^{0} ⊃ F̃^{1,0} ⊃ F̃^{1,1} ⊃ ⋯ ⊃ F̃^{1,m_{1}}

1This means that the collection of factors (G^{0}, G^{1}, …, G^{n}) is uniquely defined by E.

with stable isotype quotients F^{1,ν}/F^{1,ν+1} = Q^{0} such that Hom(F̃^{1,m_{1}}, Q^{0}) = 0. Then we put F^{1} = F̃^{1,m_{1}} and apply whole of the previous procedure to F^{1} instead of E. We get F^{2} and so on. So, each quotient G^{i} = F^{i}/F^{i+1} is filtered by F^{i,ν} = F̃^{i,ν}/F^{i+1} and this filtration has stable isotype quotients Q^{i} = F^{i,ν}/F^{i,ν+1} = F̃^{i,ν}/F̃^{i,ν+1} such that Hom(F^{i+1}, Q^{i}) = 0. This implies the orthogonality conditions (5-4).

5.3. Stability of rigid sheaves on Del Pezzo surface. We write γ_{can} and μ_{can} for the Giesecker and Mumford - Takemoto slopes w. r. t. the anticanonical divisor −K_{S} of a Del Pezzo surface S, that is

γ_{can}(E) = \frac{χ(E ⊗ ω^{∗}^{⊗n})}{rkE} ∈ ℚ[n], μ_{can}(E) = −\frac{c_{1}(E) · K_{S}}{rkE} ∈ ℚ.

The Riemann - Roch formula (4-1) :

γ_{can}(E) = 1 + \frac{1}{2} (μ_{can}(E) − μ_{can}(ω^{⊗n})) + q(E) + q(ω^{⊗n}) − \frac{c_{1}(E) · c_{1}(ω^{⊗n})}{rk(E)} =

= 1 + \frac{1}{2} nK_{S}^{2} + \frac{1}{2} n^{2}K_{S}^{2} + \frac{1}{2} μ_{can}(E) + q(E) + \frac{c_{1}(E) · K_{S}}{rkE} n

shows that μ_{can}(E) is the leading coefficient of the normalized Hilbert polynomial γ_{can}(E) among the coefficients that do depend on E. So, μ_{can}-stability implies γ_{can}-stability and γ_{can}-semistability implies μ_{can}-semistability.

5.3.1. LEMMA. Let E, F be μ_{can} - semistable sheaves on S. Then the inequality μ_{can}(E) > μ_{can}(F) − K_{S}^{2} implies Ext^{2}(F, E) = 0. Proof. By the Serre duality, Ext^{2}(F, E) ≅ Hom^{∗}(E, F (K_{s})). On the other hand,

μ_{can}(F (K_{S})) = \frac{c_{1}(F (K_{S})) · (−K_{S})}{rk(F (K_{S}))} = \frac{−c_{1}(F) · K_{S} − rk(F)K_{S}^{2}}{rk(F)} = μ_{can}(F) − K_{S}^{2} < μ_{can}(E).

Using properties of semistable sheaves (n^{∘} 5.2.1), we obtain Hom(E, F (K_{S})) = 0. □

5.3.2. COROLLARY. Let E, F be γ_{can} - semistable sheaves on S. Then the inequality γ_{can}(E) > γ_{can}(F (K_{S})) implies Ext^{2}(F, E) = 0. □

5.3.3. PROPOSITION. Locally free exceptional sheaves on S are μ_{can}-stable. Proof. Let us use here a simple trick introduced by S. Zube in [Zu1]. Namely, restrict an exceptional vector bundle E from S onto a smooth elliptic curve D ∈ | − K_{S}|. Then applying Hom(E, ∗) to the exact triple

0 ^{}E(K_{S}) ^{}E ^{}E|_{D} ^{}0

we get Hom_{D}(E|_{D}, E|_{D}) = Hom_{S}(E, E|_{D}) = ℂ, because of

Ext^{k}(E, E(K_{S}))^{∗} = Ext^{2−k}(E, E) = \{ \begin{matrix}ℂ, & for k = 2 \\ 0, & for k ≠ 2 \end{matrix} .

So, the restriction E′ = E|_{D} is a simple vector bundle over elliptic curve. By the Atiyah

theorem (see [At] and [Ku1]), it is stable w. r. t. the slope μ(E) = \frac{deg(E)}{rk(E)} on D. Since μ(E′) = μ_{can}(E), any destabilizing subsheaf for E on S would destabilize E′ over D as well. Hence, E is μ_{can}-stable on S. □

5.3.4. COROLLARY. Let E′ be restriction of an exceptional bundle E from S onto a smooth elliptic curve D ∈ | − K_{S}|. Then μ(E′) = μ_{can}(E) and E′ is stable. □

5.3.5. PROPOSITION. Let {E_{1}, E_{2}} be an exceptional pair of sheaves on S. Then at least one of the spaces Hom(E_{1}, E_{2}), Ext^{1}(E_{1}, E_{2}) is zero. Moreover, if all E_{i} are locally free, then

μ_{can}(E_{1}) = μ_{can}(E_{2}) ⟺ Ext^{1}(E_{1}, E_{2}) = 0, Hom(E_{1}, E_{2}) = 0,
μ_{can}(E_{1}) < μ_{can}(E_{2}) ⟺ Ext^{1}(E_{1}, E_{2}) = 0, Hom(E_{1}, E_{2}) ≠ 0,
μ_{can}(E_{1}) > μ_{can}(E_{2}) ⟺ Ext^{1}(E_{1}, E_{2}) ≠ 0, Hom(E_{1}, E_{2}) = 0.

Proof. By the corollary from n^{∘} 4.3.2 an exceptional sheaf on S is either locally free or a torsion sheaf � _{e}(d), where e is an exceptional curve.

Firstly, we suppose that both E_{1} and E_{2} are locally free sheaves. Consider the restriction sequence to a smooth elliptic curve D ∈ | − K_{S}|:

0 ^{}E_{1}^{∗} ⊗ E_{2}(K_{s}) ^{}E_{1}^{∗} ⊗ E_{2}  ^{}(E_{1}^{∗} ⊗ E_{2})|_{D}  ^{}0 .

Combining the Serre duality and the definition of exceptional pair, we get

Ext^{q}(E_{1}, E_{2}(K_{S}))^{∗} ≅ Ext^{2−q}(E_{2}, E_{1}) = 0 ∀q .

Hence, the restriction sequence induces the isomorphism

Ext_{S}^{q}(E_{1}, E_{2}) ≅ Ext_{e}^{q}(E_{1}|_{D}, E_{2}|_{D}) .

The previous corollary implies that E_{i}|_{D} is a stable bundle on the elliptic curve D and μ(E_{i}|_{D}) = μ_{can}(E_{i}). Therefore, Ext^{1}(E_{1}, E_{2}) = 0 whenever μ_{can}(E_{1}) < μ(E_{2}). In the converse case, Hom(E_{1}, E_{2}) = 0.

In the case μ_{can}(E_{1}) = μ_{can}(E_{2}) any nonzero map E_{1} → E_{2} is isomorphism, whence Hom(E_{1}, E_{2}) = 0. In additional, Riemann-Roch formula (4-1) gives that χ(E_{1}, E_{2}) = 0. This means that Ext^{1}(E_{1}, E_{2}) = 0 as well.

The Proposition holds if E_{2} is a locally free sheaf and E_{1} =� _{e}(d), since any map from a torsion sheaf to a locally free sheaf is zero.

Now, we assume that E_{1} =� _{e_{1}}(d_{1}) and E_{2} =� _{e_{2}}(d_{2}). If e_{1} ≠ e_{2}, then Ext^{q}(E_{1}, E_{2}) = 0 ∀q. If e_{1} = e_{2} = e, then the Proposition follows from the equality:

Ext_{S}^{q}(E_{1}, E_{2}) = Ext_{e}^{q}(� _{e}(d_{1}),� _{e}(d_{2})).

Finally, suppose that E_{1} = E is a locally free sheaf and E_{2} =� _{e}(d). We have:

Ext_{S}^{q}(E_{1}, E_{2}) = Ext_{e}^{q}(E|_{e},� _{e}(d)) .

Besides, by the Serre duality and by the definition of exceptional pair, for any q we obtain

0 = Ext_{S}^{q}(E_{2}, E_{1})^{∗} ≅ Ext_{S}^{2−q}(E_{1}, E_{2}(K_{S})) ≅ Ext_{e}^{2−q}(E|_{e},� _{e}(d − 1))

(in the last equality we use the relation: eK_{S} = −1). In particular, Hom_{e}(E|_{e},� _{e}(d − 1)) = Ext_{e}^{1}(E|_{e},� _{e}(d − 1)) = 0. Now we apply the functor Ext_{e}(E|_{e}, ·) to the restriction sequence

0 ^{}� _{e}(d − 1) ^{}� _{e}(d) ^{}� _{x} ^{}0

and get

Ext_{S}^{1}(E_{1}, E_{2}) ≅ Ext_{e}^{1}(E|_{e},� _{e}(d)) ≅ Ext^{1}(E|_{e},� _{x}) = 0.
□

5.4. Spectral sequences. More precise results than the above theorem may be obtained by a straightforward computations of Ext^{p+q}(E, E) via spectral sequences associated with the Harder - Narasimhan and Jordan - Holder filtrations and using estimations from n^{∘} 4.2. Just to illustrate this approach, let us prove the μ_{can}-semistability of exceptional E without the above ‘elliptic trick’.

We have to show that the Harder - Narasimhan filtration E = F^{0} ⊃ F^{1} ⋯ ⊃ F^{n+1} = 0 , for E, has just one term. Let Gr^{i} = F^{i}/F^{i+1} be its quotients. Consider the spectral sequence with

E_{1}^{p,q} = ⊕_{β−α=p} Ext^{p+q}(Gr^{α}, Gr^{β})

which converges to Ext^{p+q}(E, E). It looks like in fig. D on page 39, because of

Ext^{2}(Gr^{β}, Gr^{α}) = Hom(Gr^{α}, Grβ) = 0 for α > β

by the semistability of Gr^{ν} and the lemma from n^{∘} 4.2.6. So, E_{2} = E_{∞} and E_{1}^{(0,1)} = E_{∞}^{(0,1)} = 0, that is, all Gr^{ν} are rigid, the differential

d^{(−1,2)} : ⊕Hom ( Gr^{ν}, Gr^{ν−1} ) ^{}⊕ Ext^{2} ( Gr^{ν}, Gr^{ν} )

has to be an isomorphism, and the differential

d^{(0,0)} : ⊕Hom ( Gr^{ν}, Gr^{ν} ) ^{}⊕ Ext^{2} ( Gr^{ν}, Gr^{ν+1} )

has to be an epimorphism with 1-dimensional kernel. This implies that there is at least one non zero Gr^{ν} exactly like in the proof of the theorem from n^{∘} 4.3.3. Quite similar strategy allows to prove the following

5.4.1. LEMMA. Suppose that a γ_{can}-semistable rigid sheaf F has a filtration

F = F^{1} ⊃ F^{2} ⊃ ⋯ ⊃ F^{k} ⊃ F^{k+1} = 0

with stable isotype quotients E = F^{i}/F^{i+1} (the same for all i). Then E is exceptional and F = k E in a sense that

F = \underbrace{E ⊕ E ⊕ ⋯ ⊕ E}_{k} .

Proof. Since E is stable, Hom(E, E) = ℂ and by the corollary from n^{∘} 5.3.2 we have Ext^{2}(E, E) = 0. So,

E_{1}^{p,q} = ⊕_{β−α=p} Ext^{p+q}(Gr^{α}, Gr^{β})

which computes Ext^{p+q}(E, E), looks like in fig. E on page 40.

The topmost left term E_{1}^{−k,k+1} = Ext^{1}(E, E) fails into E_{∞}^{1} = Ext^{1}(F, F) = 0. Hence E is rigid, that is, exceptional. □

[figure: q; p; d; ∗; ∗; 0; ∗; ∗; 0; ∗; ∗; 0; ∗; ∗; ∗; 0; ∗; ∗; 0; ∗; ∗; 0; ∗; ∗]

Fig. D. E_{1}^{p,q} = ⊕_{β−α=p} Ext^{p+q}(Gr^{α}, Gr^{β})

5.5. Structure of rigid sheaves. In this section we apply the above technique to prove that any torsion free rigid sheaf E on a Del Pezzo surface S is a direct sum of exceptional locally free sheaves. The proof is nothing but a step by step analysis of the spectral sequences associated with Jordan - Holder and Harder - Narasimhan filtrations.

5.5.1. LEMMA. A γ_{can}-semistable rigid sheaf E on a Del Pezzo surface is a direct sum of exceptional locally free sheaves. Proof. Let E = F^{0} ⊃ F^{1} ⊃ ⋯ ⊃ F^{n} ⊃ F^{n+1} = 0 be the isotype Jordan - Holder filtration constructed in n^{∘} 5.2.3. Each exact triple

0 ^{}F^{i+1}  ^{}F^{i}  ^{}G^{i}  ^{}0

satisfies the Mukai lemma n^{∘} 4.2.2, that is, Hom(F^{i+1}, G^{i}) = Ext^{2}(G^{i}, F^{i+1}) = 0, because of the filtration properties and the corollary from n^{∘} 5.3.2. Applying the Mukai lemma inductively, we get for all i

Ext^{1}(F^{i}, F^{i}) = Ext^{1}(G^{i}, G^{i}) = 0.  (5-6)

So, each isotype quotient G^{i} is rigid. The previous lemma (see n^{∘} 5.4.1) shows that G_{i} is a multiplicity of some exceptional vector bundle E_{i}:

G^{i} = k_{i}E_{i} = \underbrace{E_{i} ⊕ E_{i} ⊕ ⋯ ⊕ E_{i}}_{k_{i}} .

The corollary from n^{∘} 5.3.2 and properties of isotype Jordan - Holder filtration filtration, show that Hom(G^{i+1}, G^{i}) = Hom(G^{i}, G^{i+1}) = 0 and Ext^{2}(G^{i}, G^{j}) = 0 for all i, j. Therefore,

E_{1}^{p,q} = ⊕_{β−α=p} Ext^{p+q}(Gr^{α}, Gr^{β})

[figure: q; p; d; 0; ∗; ∗; 0; ∗; ∗; 0; ∗; ∗; 0; ∗; ∗; 0; ∗; ∗; 0; ∗; ∗; 0; ∗; ∗]
Fig. E. E_{1}^{p,q} = ⊕_{β−α=p} Ext^{p+q}(Gr^{α}, Gr^{β}) for the lemma from n^{∘} 5.4.1

which computes Ext^{p+q}(E, E), looks like in fig. F on page 41.

Again, the topmost left term E_{1}^{−n,n+1} = Ext^{1}(G^{n}, G^{0}) ⊂ E_{∞}^{1} = Ext^{1}(E, E) = 0. Hence, the exceptional locally free sheaves E_{0} and E_{n} satisfy Ext^{1}(E_{n}, E_{0}) = 0. Since they are stable and have equal slopes, we also have either E_{0} = E_{n} or Hom(E_{0}, E_{n}) = Hom(E_{n}, E_{0}) = 0, and anyway, Ext^{2}(E_{0}, E_{n}) = Ext^{2}(E_{n}, E_{0}) = 0. The coincidence μ_{can}(E_{0}) = μ_{can}(E_{n}) also means that ‘the skew Euler characteristic’ χ(E_{n}, E_{0}) − χ(E_{0}, E_{n}) vanishes by the Riemann - Roch formula (n^{∘} 4.1.1). So, Ext^{1}(E_{0}, E_{n}) = Ext^{1}(E_{n}, E_{0}) = 0 and Ext^{1}(G^{0}, G^{n}) = 0 as well.

Now we can prove that E is a direct sum of its isotype quotients G^{i}’s using the induction on the filtration length n. Case n = 1 was just considered above. Further, in the exact triple

0 ^{}F^{1}  ^{}E ^{}G^{0}  ^{}0

F^{1} is rigid, semistable, and decomposed as F^{1} = ⊕_{i=1}^{n} G^{i} by the inductive assertion. Since

Ext^{1}(G^{0}, G^{n}) = 0, we have the splitting E = E′ ⊕ G^{n} with γ_{can}-semistable rigid E′ whose

isotype Jordan - Holder filtration has smaller length, i. e. E′ = ⊕_{i=0}^{n−1} G^{i} by induction. □

5.5.2. THEOREM. Any torsion free rigid sheaf E on a Del Pezzo surface S is a direct sum of exceptional locally free sheaves. Proof. Consider a spectral sequence that computes Ext^{p+q}(E, E) using the filtration of Harder - Narasimhan (see n^{∘} 5.2.2):

E = F^{0} ⊃ F^{1} ⊃ ⋯ ⊃ F^{n} ⊃ F^{n+1} = 0

whose quotients Gr^{i} = F^{i}/F^{i+1} are semistable and have strictly increasing slopes γ_{can}(Gr^{i}) < γ_{can}(Gr^{i+1}). The properties of semistable sheaves (see n^{∘} 5.2.1, n^{∘} 5.3.2) imply that

Hom(Gr^{i}, Gr^{j}) = 0 when i > j and Ext^{2}(Gr^{i}, Gr^{j}) = 0 whenever i ≤ j.  (5-7)

[figure: q; p; d; 0; ∗; ∗; 0; ∗; ∗; 0; ∗; 0; 0; ∗; ∗; 0; ∗; 0; 0; ∗; ∗; 0; ∗; ∗]
Fig. F. E_{1}^{p,q} = ⊕_{β−α=p} Ext^{p+q}(Gr^{α}, Gr^{β}) for the lemma from n^{∘} 5.5.1

Therefore,

E_{1}^{p,q} = ⊕_{β−α=p} Ext^{p+q}(Gr^{α}, Gr^{β})

looks like in fig. G on page 42.

First of all, we see that ⊕Ext^{1}(Gr^{i}, Gr^{i}) = E_{1}^{0,1} = E_{∞}^{0,1} ⊂ Ext^{1}(E, E) = 0, that is, each quotient Gr^{i} is rigid and (by the lemma from n^{∘} 5.5.1) splits as Gr^{i} = ⊕_{α} E_{i}^{α}, where E_{i}^{α} are exceptional and have γ_{can}(E_{i}^{α}) = γ_{can}(Gr^{i}).

Secondary, we see that ⊕Ext^{1}(Gr^{i}, Gr^{i−1}) = E_{1}^{−1,2} = E_{∞}^{−1,2} ⊂ Ext^{1}(E, E) = 0, that is,

Ext^{1}(Gr^{i}, Gr^{i−1}) = 0 ∀ i.  (5-8)

Let us show that Ext^{1}(Gr^{i−1}, Gr^{i}) = 0 ∀ i as well.

For this, it is enough to check that Ext^{1}(E_{i−1}^{α}, E_{i}^{β}) = 0 ∀ i, α, β. Since γ_{can}(E_{i−1}^{α}) = γ_{can}(Gr^{i−1}) < γ_{can}(Gr^{i}) = γ_{can}(E_{i}^{β}), there are two possibilities: either μ_{can}(E_{i−1}^{α}) = μ_{can}(E_{i}^{β}) or μ_{can}(E_{i−1}^{α}) < μ_{can}(E_{i}^{β}). In the first case, the skewsymmetric part of the Euler characteristic χ(E_{i−1}^{α}, E_{i}^{β}) vanishes and we use the same arguments as in the proof of the lemma from n^{∘} 5.5.1. In the second case, let us restrict the bundle (E_{i}^{β})^{∗} ⊗ E_{i−1}^{α} to an elliptic curve C ∈ | − K_{S}| and consider a long exact sequence of cohomologies

⋯ ^{}Hom(E_{i}^{β}, E_{i−1}^{α}) ^{}Hom(E_{i}^{β}|_{C}, E_{i−1}^{α}|_{C}) ^{} Ext^{1}(E_{i}^{β}, E_{i−1}^{α}(K_{S})) ^{}Ext^{1}(E_{i}^{β}, E_{i−1}^{α}) ^{}⋯

which comes from the exact triple

0 ^{}(E_{i}^{β})^{∗} ⊗ E_{i−1}^{α} ⊗ K_{S}  ^{}(E_{i}^{β})^{∗} ⊗ E_{i−1}^{α}  ^{}(E_{i}^{β})^{∗} ⊗ E_{i−1}^{α}|_{C}  ^{}0.

[figure: q; p; d; ∗; ∗; 0; ∗; ∗; 0; ∗; ∗; 0; 0; ∗; ∗; 0; ∗; 0; 0; ∗; ∗; 0; ∗; ∗]
Fig. G. E_{1}^{p,q} = ⊕_{β−α=p} Ext^{p+q}(Gr^{α}, Gr^{β}) for the theorem from n^{∘} 5.5.2

Since exceptional bundles are stabile, the corollary from n^{∘} 5.3.4 implies that

Hom(E_{i}^{β}, E_{i−1}^{α}) = Hom(E_{i}^{β}|_{C}, E_{i−1}^{α}|_{C}) = 0.

On the other side, (5-8) implies Ext^{1}(E_{i}^{β}, E_{i−1}^{α}) = 0. So, we get Ext^{1} ( E_{i}^{β} , E_{i−1}^{α}(K_{S}) ) = 0,

or by the Serre duality, Ext^{1} ( E_{i−1}^{α}, E_{i}^{β} ) = 0 as required.

Now we can prove that E is a direct sum of its Harder - Narasimhan quotients inductively on the length n of the Harder - Narasimhan filtration. The case n = 1 is already done. In general case, consider the first Harder - Narasimhan quotient:

0 ^{}F^{1}  ^{}E ^{}Gr^{1}  ^{}0.

Semistablity of Gr^{i}’s and the vanishing conditions (5-7) show that

Hom(F^{1}, Gr^{1}) = Ext^{2}(Gr^{1}, F^{1}) = 0.

Hence, F^{1} is rigid by the Mukai lemma from n^{∘} 4.2.2 and we can suppose inductively that F^{1} is a direct sum of its Harder - Narasimhan quotients: F^{1} = ⊕_{i=2}^{n} Gr^{i}. Since Ext^{1}(Gr^{1}, Gr^{2}) = 0,

the sheaf Gr^{2} is a direct summand of E, that is, E = E′ ⊕Gr^{2} and we can apply the inductive assumption to E′. Now to finish the prove, it remains to apply the lemma from n^{∘} 5.5.1 to the Harder - Narasimhan quotients of E. □

5.5.3. COROLLARY. Let F be an extra rigid locally free sheaf on S. Then there exists an exceptional collection (F_{1}, F_{2}, …, F_{r}) such that F = ⊕_{i=1}^{r} y_{i}F_{i}.

Proof. Since F is extra rigid, by the previous theorem, F = ⊕_{i=1}^{r} y_{i}F_{i} and

Ext^{>0}(F_{i}, F_{j}) = 0 ∀ i, j .

§ 6. Del Pezzo surfaces: transitivity of mutations. 43

So, if we renumber the bundles F_{i} in such a way that μ_{can}(F_{i}) ≤ μ_{can}(F_{i+1}), then they will form an exceptional collection, because Hom(F_{j}, F_{i}) = 0 will hold for i < j by the proposition from n^{∘} 5.3.5. □

5.5.4. Remark. If an extra rigid locally free sheaf F belongs to a subcategory [E_{i}, …, E_{j}] generated by some subcollection of an exceptional base {E_{1}, E_{2}, …, E_{m}} for � _{�oh}^{b} (S), then all exceptional direct summands F_{ν} in the decomposition F = ⊕_{ν} y_{ν}F_{ν} have to belong to

[E_{i}, …, E_{j}] as well. Indeed, taking an other helix foundation generating � _{�oh}^{b} (S), we can suppose that i = 1 and j = k, say. That is, F ∈ Ob [E_{1} … E_{k}] = Ob [E_{k+1} … E_{m}]^{⊥}. But

F ∈ Ob [E_{k+1} … E_{m}]^{⊥} ⟹ all F_{ν} ∈ Ob [E_{k+1} … E_{m}]^{⊥}

certainly.

§6. Del Pezzo surfaces: transitivity of mutations.

In this paragraph we prove the theorem from n^{∘} 4.6.1 for Del Pezzo surfaces S_{d}.

6.1. Terminology concerned with the braid group action. We call two exceptional collections equivalent, if they lay in the same braid group orbit. We say that an exceptional collection of sheaves {F_{1}, F_{2}, …, F_{k}} is obtained from an exceptional collection {E_{1}, E_{2}, …, E_{m}} by mutations, if k ≤ m and the collection {F_{1}, F_{2}, …, F_{k}} can be extended to some bigger exceptional collection {F_{1}, F_{2}, …, F_{m}} equivalent to {E_{1}, E_{2}, …, E_{m}}. An exceptional collection is called constructible, if it is obtained by mutations from the initial collection given in the theorem n^{∘} 4.1.2:

{� _{e_{1}}(−1), …, , � _{e_{d}}(−1), � _{S_{d}}, � _{S_{d}}(h), � _{S_{d}}(2h)}

where h and e_{1}, e_{2}, …, e_{d} are, accordingly, the proper preimages of a line and consequently blow-up’ed points on ℙ_{2}.

It follows from n^{∘} 2.8.1 that a full exceptional collection {E_{1}, E_{2}, …, E_{k}} is equivalent to {E_{n}(K_{S_{d}}), E_{1}, …, E_{n−1}}, because of E_{n}(K_{S_{d}}) = L_{E_{1}}L_{E_{2}} ⋯ L_{E_{n−1}}E_{n}. In particular, shifting a helix foundation by the helix period, we get an equivalent collection, i. e. full exceptional collections

{E_{1}, E_{2}, …, E_{k}} and {E_{1}(xK_{S_{d}}), …, E_{k}(xK_{S_{d}})}  (6-1)

are equivalent for all x ∈ ℤ. The theorem from n^{∘} 4.6.1 can be reformulated now as

6.1.1. THEOREM. Any exceptional collection on S_{d} is constructible.

6.2. Proof of the theorem.. We will use an induction on d. For d = 0 we have S_{d} = ℙ^{2}. Historically, this case was essentially done by J.-M. Drézet and J. Le Potier in [DLP] two years before the helix theory birth. This was the first appearance of exceptional vector bundles (see also [Dr1]). The helix theory viewpoint on the description of exceptional bundles and helices on ℙ_{2} is based on the arithmetical properties of Mukai lattice K_{0}(ℙ_{2}) and will discuss it in details little bit later, in n^{∘} 7.2.

The rest of this paragraph will be devoted to the inductive step. Let us assume that d > 0 and the theorem holds for S_{d−1}. We will prove now that the theorem holds for S_{d}. We identify � _{�oh}^{b} (S_{d−1}) with the left orthogonal ^{⊥}� _{e_{d}}(−1) ⊂ � _{�oh}^{b} (S_{d}) (see proof of the theorem from n^{∘} 4.1.2) via pull back under the blow up morphism S_{d} [figure] S_{d−1} with the exceptional divisor e_{d}. Then inductive hypothesis says that all exceptional collections laying in ^{⊥}� _{e_{d}}(−1) can be obtained from the standard one. So, we have to mute an arbitrary exceptional collection to such one that contains � _{e_{d}}(−1) and has other elements from ^{⊥}� _{e_{d}}(−1). This will be done in two steps.

6.2.1. The first step is a kind of ‘integer simplex-method algorithm’ modified to deal with objects of an Abelian category. We formulate it as

6.2.2. PROPOSITION. Let {E_{0}, E_{1}, …, E_{m}}, {F_{0}, F_{1}, …, F_{k}} be two exceptional collection on S_{d} such that all F_{i} and all E_{j} except for E_{0} are locally free but E_{0} is either locally free or equal � _{e_{d}}(−1). Suppose that {E_{0}, E_{1}, …, E_{m}} can be extended to some exceptional base

for � _{�oh}^{b} (S_{d}) and for some positive integer multiplicities α_{ν}, x_{μ} both the sums F = ⊕_{i=0}^{k} α_{i}F_{i},

E = ⊕_{i=1}^{m} x_{i} E_{i} are extra rigid and fit into an exact triple

0 ^{}⊕_{i=1}^{m} x_{i} E_{i}  ^{}F ^{}x_{0} E_{0}  ^{}0 .  (6-2)

Then ∑_{i=0}^{k} rkF_{i} ≥ ∑_{i=0}^{m} rkE_{i} and {F_{0}, F_{1}, …, F_{k}} is obtained from {E_{0}, E_{1}, …, E_{m}} by muta-

tions (in particular, k ≤ m). Moreover, if m = 1 and E_{0} is locally free, then the equality

∑_{i=0}^{k} rkF_{i} = ∑_{i=0}^{1} rkE_{i}

implies that either k = 1 and F_{0} = E_{0}, F_{1} = E_{1}, or k = 0.

The proof consequently modifies ‘the pivot’ {E_{0}, E_{1}, …, E_{m}} via simplex-method using arithmetics from n^{∘} 1.7 and induction on m. We give it in n^{∘} 6.3 below.

6.2.3. The second step reduces the general case of the theorem to the exceptional collections {F_{0}, F_{1}, …, F_{k}} satisfying the previous proposition. Namely, in n^{∘} 6.4 we will prove

6.2.4. PROPOSITION. Any exceptional collection on S_{d} can be obtained from an excep-

tional collection {F_{0}, F_{1}, …, F_{k}} such that F = ⊕_{i=0}^{k} F_{i} is an extra rigid locally free sheaf

for which there is an exceptional collection {� _{e_{d}}(−1), E_{1}, …, E_{m}} such that the canonical morphism F ^{can}^{}Hom^{∗}(F,� _{e_{d}}(−1)) ⊗� _{e_{d}}(−1) is surjective and fits into exact triple:

0 ^{}⊕_{i=1}^{m} x_{i} E_{i} ^{}F ^{can}^{}x_{0}� _{e_{d}}(−1), ^{}0

where ⊕_{i=1}^{m} x_{i} E_{i} is extra rigid and locally free as well.

6.2.5. Final part: use the inductive hypothezis. Two previous propositions show that

any exceptional collection on S_{d} can be obtained from some exceptional collection of the form {� _{e_{d}}(−1), E_{1}, …, E_{m}}. Since all the bundles E_{i} are orthogonal to� _{e_{d}}(−1), the exceptional collection {E_{1}, E_{2}, …, E_{m}} is a pull-back of some exceptional collection on S_{d−1} under the blow up S_{d} ^{σ_{d}} [figure] S_{d−1} (comp. with the proof of the theorem from n^{∘} 4.1.2). So, by the inductive hypothesis, the collection {E_{1}, E_{2}, …, E_{m}} is constructible inside the category ^{⊥}� _{e_{d}}(−1), i. e. the whole ‘pivot’ {� _{e_{d}}(−1), E_{1}, …, E_{m}} is constructible on S_{d}. This finishes the proof of the main theorem n^{∘} 6.1.1.

So, it remains to prove the propositions from n^{∘} 6.2.2 and n^{∘} 6.2.4.

6.3. Proof of the proposition n^{∘} 6.2.2. Let us use an induction on m and consider firstly the base case m = 1.

6.3.1. Base of induction. For m = 1 the sequence (6-2) turns to exact triple

0 ^{}x_{1} E_{1}  ^{}F ^{}x_{0} E_{0}  ^{}0 .  (6-3)

Since the pair {E_{0}, E_{1}} is assumed extendable to some exceptional base for � _{�oh}^{b} (S_{d}), all exceptional summands F_{i}, of F, have to belong the subcategory [E_{0}, E_{1}] (see n^{∘} 5.5.4).

If Ext^{1}(E_{0}, E_{1}) = 0, then F = x_{0} E_{0} ⊕ x_{1} E_{1} and there is nothing to prove.

If dim Ext^{1}(E_{0}, E_{1}) = h > 0, then the description of exceptional objects in the Mukai lattice K_{0}([E_{0}, E_{1}]) (see n^{∘} 1.7) shows that F contains at most two direct summands F_{0}, F_{1}, which form an exceptional pair obtained from {E_{0}, E_{1}} by mutations, i.e., F = α_{0}F_{0} ⊕ α_{1}F_{1} with α_{0} > 0, α_{1} ≥ 0. If we write the class of F_{i} in K_{0}([E_{0}, E_{1}]) as a_{i}e_{0} + b_{i}e_{1}, where e_{i} is the class of E_{i}, then

1 = a_{i}^{2} + b_{1}^{2} − ha_{i}b_{i}

implies that the integers a_{i}, b_{i} have the same sign. So, if the both rkE_{i} > 0, then (if α_{1} > 0) the equality rkF_{0} + rkF_{1} = rkE_{0} + rkE_{1} is possible only for E_{0} = F_{0} and E_{1} = F_{1}. The case m = 1 is completely proved.

6.3.2. The inductive step consists of appropriate modification of the both collections E, F. This modification splits into a series of elementary transformations of two different kinds described in two lemmas below.

6.3.3. LEMMA. Let F be an extra rigid sheaf and {E_{0}, E_{1}, …, E_{m}} be an exceptional collection. Then an exact sequence

0 ^{}⊕_{i=1}^{m} x_{i} E_{i}  ^{}F ^{}x_{0} E_{0}  ^{}0  (6-4)

produces a commutative diagram

[figure: x_{1} E_{1}; ⊂; E′; x_{0} E_{0}; ⊕_{i=1}^{m} x_{i} E_{i}; ⊂; F; x_{0} E_{0}; ‖; ⊕_{i=2}^{m} x_{i} E_{i}; ∪; {= \atop =}; ⊕_{i=2}^{m} x_{i} E_{i}; ∪],  (6-5)

where E′ is an extra rigid sheaf such that Ext^{q}( ⊕_{i=2}^{m} x_{i} E_{i}, E′) = 0 ∀q.

Proof. The diagram is induced by its left column and the middle row. Required orthogonality: Ext^{q}( ⊕_{i=2}^{m} x_{i} E_{i}, E′) = 0 ∀q follows from the exceptionality of the collection

{E_{0}, E_{1}, …, E_{m}} after applying the functor Ext( ⊕_{i=2}^{m} x_{i}E_{i}, ∗) to the first row of the diagramm.

Further, applying Ext( ∗ , E_{0}) to the middle row, we get Ext^{>0}(F, E_{0}) = 0, and then, applying to the same row Ext(F , ∗) and using the extra rigidness: Ext^{>0}(F, F) = 0, we get Ext^{2}(F, ⊕_{i=1}^{m} x_{i} E_{i}) = 0. Now, taking Ext(F , ∗) in the middle column, we obtain the vanishing:

Ext^{>0}(F, E′) = 0. Finally, applying to the middle column Ext( ∗ , E′), we get the required extra rigidness: Ext^{>0}(E′, E′) = 0. □

6.3.4. LEMMA. Let F be an extra rigid sheaf and {E_{0}, E_{1}, …, E_{m}} be an exceptional collection. Then an exact sequence

0 ^{}⊕_{i=2}^{m} x_{i} E_{i}  ^{}F ^{}x_{0} E_{0} ⊕ x_{1} E_{1}  ^{}0  (6-6)

produces a commutative diagram

[figure: x_{0} E_{0} = = = = = = x_{0} E_{0}; ⊕_{i=2}^{m} x_{i} E_{i}; ⊂; F; x_{0} E_{0} ⊕ x_{1} E_{1}; ⊕_{i=2}^{m} x_{i} E_{i}; ‖; ⊂; E′′; ∪; x_{1} E_{1}; ∪]  (6-7)

where E′′ is an extra rigid sheaf such that Ext^{q}(E′′, x_{0} E_{0}) = 0 ∀q.

Proof. Completely similar to the previous one. □

6.3.5. Reduction to the case of locally free E_{0}. Let E_{0} = � _{e_{d}}(−1). We are going

to mute the collection {E_{0}, E_{1}, …, E_{m}} into a new exceptional collection {E′_{0}, E′_{1}, …, E′_{m′}}, which also satisfies the proposition, i. e. represents F by the exact triple

0 ^{}⊕_{i=1}^{m′} x′_{i} E′_{i}  ^{}F ^{}x′_{0} E′_{0}  ^{}0 ,  (6-8)

but has a locally free E′_{0}. To this aim, consider two exact sequences given by the lemma from n^{∘} 6.3.3:

0 ^{}⊕_{i=2}^{m} x_{i} E_{i}  ^{}F ^{}E′  ^{}0,
0 ^{}x_{1} E_{1}  ^{}E′  ^{}x_{0}� _{e_{d}}(−1) ^{}0.

Let T E′ be the torsion sheaf of E′. Since E_{1} is locally free, Hom(T E′, x_{1} E_{1}) = 0 and we have the following commutative diagram

[figure: x_{1} E_{1}; ⊂; E′/T E′; G; x_{1} E_{1}; ‖; ⊂; E′; x_{0}� _{e_{d}}(−1); T E′; ∪; = = = = = = =; T E′; ∪] ,  (6-9)

where G has to be of the form G = ⊕_{i} β_{i}� _{e_{d}}(r_{i}) ⊕ G′ with dim Supp G′ = 0. Since E_{1} is

locally free, we have Ext^{1}(G′, x_{1} E_{1}) = 0, i. e. G′ is a direct summand of the torsion free sheaf E′/T E′. Hence, G′ = 0 and G = ⊕_{i} β_{i}� _{e_{d}}(r_{i}), where all r_{i} ≥ −1, certainly. Let us show that all r_{i} ≤ −1 as well.

Indeed, since E′/T E′ has no torsion, each � _{e_{d}}(r_{i}) is non trivially extended by E_{1} in the first row of the diagram, that is, Ext_{S_{d}}^{1}(� _{e_{d}}(r_{i}), E_{1}) ≠ 0 ∀i. On the other hand, the pair {� _{e_{d}}(−1), E_{1}} is exceptional, i. e. ∀q we have

0 = Ext_{S_{d}}^{q}(E_{1},� _{e_{d}}(−1)) = Ext_{e_{d}}^{q}(E_{1}|_{e_{d}},� _{e_{d}}(−1)) .

Therefore, E_{1}|_{e_{d}} = α� _{e_{d}} and

Ext_{S_{d}}^{1}(� _{e_{d}}(r_{i}), E_{1})^{∗} = Ext_{S_{d}}^{1}(E_{1},� _{e_{d}}(r_{i}) ⊗ K_{S_{d}}) = Ext_{e_{d}}^{1}(E_{1}|_{e_{d}},� _{e_{d}}(r_{i} − 1)) = = Ext_{e_{d}}^{1}(α� _{e_{d}},� _{e_{d}}(r_{i} − 1)) ≠ 0

Thus, all r_{i} ≤ −1 and G = β� _{e_{d}}(−1). So, the first column of the diagram (6-9) splits and we can write

E′ = E′/T E′ ⊕ T E′ = E′/T E′ ⊕ y_{2}� _{e_{d}}(−1) .

Here E′/T E′ is locally free and extra rigid (because E′ is) and fits into exact sequence

0 ^{}x_{1} E_{1}  ^{}E′/T E′  ^{}(x_{0} − y_{2})� _{e_{d}}(−1) \overline{}0

Hence (see n^{∘} 5.5.4 and n^{∘} 1.7), E′/T E′ = y_{0} E′_{0} ⊕ y_{1} E′_{1} for some exceptional pair of locally free sheaves {E′_{0}, E′_{1}} obtained from the pair {� _{e_{d}}(−1), E_{1}} by mutations. Since the sheaf

E′ = y_{0} E′_{0} ⊕ y_{1} E′_{1} ⊕ y_{2}� _{e_{d}}(−1) ∈ Ob [� _{e_{d}}(−1), E_{1}]

is extra rigid, we have either y_{1} = 0, or y_{2} = 0, or y_{1} = y_{2} = 0. Anyway, E′ = y_{0} E′_{0} ⊕ y Ẽ, where {E′_{0}, Ẽ} is an exceptional pair obtained from the pair {� _{e_{d}}(−1), E_{1}} by mutations.

Now, let us note that we have the exact sequence

0 ^{}⊕_{i=2}^{m} x_{i} E_{i}  ^{}F ^{}y_{0} E′_{0} ⊕ y Ẽ ^{}0  (6-10)

for the exceptional collection (E′_{0}, Ẽ, E_{2}, … E_{m}) obtained from {E_{0}, E_{1}, …, E_{m}} by mutations. So, if y = 0, we have done. If y > 0, we apply to (6-10) the lemma from n^{∘} 6.3.4 and get exact sequences:

0 ^{}E′′  ^{}F ^{}y_{0} E′_{0}  ^{}0,
0 ^{}⊕_{i=2}^{m} x_{i} E_{i}  ^{}E′′  ^{}yẼ ^{}0

with extra rigid E′′ such that Ext_{S_{d}}^{q}(E′_{0}, E′′) = 0 ∀q. Moreover, E′′ ⊂ F has no torsion, because F is locally free. By the corollary from n^{∘} 4.3.2, E′′ is locally free, that is (see n^{∘} 5.5.4)

E′′ = ⊕_{i=1}^{m′} x′_{i} E′_{i}

for some exceptional collection of locally free sheaves {E′_{0}, E′_{1}, …, E′_{m′}}. By the inductive hypothesis, {E′_{1}, E′_{2}, …, E′_{m′}} is obtained from {Ẽ, E_{2}, …, E_{m}} by mutations. Hence, {E′_{0}, E′_{1}, …, E′_{m′}} is obtained from {E_{1}, E_{2}, …, E_{m}} as well and we have completely reduced the general case of the proposition n^{∘} 6.2.2 to the situation, where E_{0} is locally free.

6.3.6. Proof of the proposition n^{∘} 6.2.2 for locally free E_{0}. Again, let us apply the

lemma from n^{∘} 6.3.3 to the triple (6-8). We get two exact triples

0 ^{}⊕_{i=2}^{m} x′_{i} E′_{i}  ^{}F ^{}E^{(1)}  ^{}0,  (6-11)
0 ^{}x′_{1} E′_{1}  ^{}E^{(1)}  ^{}y_{0} E′_{0}  ^{}0.  (6-12)

Since E′_{1} and E′_{0} are locally free, the extra rigid sheaf E^{(1)} is locally free as well. Remark n^{∘} 5.5.4 and the inductive hypothesis imply that E^{(1)} = x_{0}^{(1)} E_{0}^{(1)}⊕x_{1}^{(1)} E_{1}^{(1)} for some exceptional pair {E_{0}^{(1)}, E_{1}^{(1)}}, which is obtained from {E′_{0}, E′_{1}} by mutations and satisfies

rkE_{0}^{(1)} + rkE_{1}^{(1)} ≥ rkE′_{0} + rkE′_{1}.  (6-13)

So, {E_{0}^{(1)}, E_{1}^{(1)}, E′_{2}, …, E′_{m}} is an exceptional collection obtained from {E′_{0}, E′_{1}, …, E′_{m}} by mutations and satisfying

rkE_{0}^{(1)} + rkE_{1}^{(1)} + ∑_{i=2}^{m} rkE′_{i} ≥ ∑_{i=0}^{m} rkE′_{i}.

By the inductive hypothesis, we can assume that x_{0}^{(1)} ≠ 0 and x_{1}^{(1)} ≠ 0. Applying the lemma from n^{∘} 6.3.4, we get two next exact sequences:

0 ^{}E^{(2)}  ^{}F ^{}x_{0}^{(1)} E_{0}^{(1)}  ^{}0
0 ^{}⊕_{i=2}^{m} x′_{i} E′_{i}  ^{}E^{(2)}  ^{}x_{1}^{(1)} E_{1}^{(1)}  ^{}0

with an extra rigid locally free E^{(2)}. Again, the inductive hypothesis and remark n^{∘} 5.5.4 show that E^{(2)} = ⊕_{i=1}^{m_{2}} x_{i}^{(2)} E_{i}^{(2)}, where {E_{1}^{(2)}, E_{2}^{(2)}, …, E_{m_{2}}^{(2)}} is an exceptional collection of locally free

sheaves obtained from {E_{1}^{(1)}, E′_{2}, …, E′_{m}} and satisfying the inequality

∑_{i=1}^{m_{2}} rkE_{i}^{(2)} ≥ rkE_{1}^{(2)} + ∑_{i=2}^{m} rkE′_{i} .  (6-14)

Thus, the extra rigid bundle F is included into

0 ^{}⊕_{i=2}^{m_{2}} x_{i}^{(2)} E_{i}^{(2)}  ^{}F ^{}x_{0}^{(2)} E_{0}^{(2)}  ^{}0 ,

where x_{0}^{(2)} E_{0}^{(2)} = x_{0}^{(1)} E_{0}^{(1)}, the collection {E_{0}^{(2)}, E_{1}^{(2)}, …, E_{m_{2}}^{(2)}} is obtained from the collection

{E′_{0}, E′_{1}, …, E′_{m}}, and ∑_{i=0}^{m_{2}} rkE_{i}^{(2)} ≥ ∑_{i=0}^{m} rkE′_{i}.

Repeating this two-step procedure, we construct for each j = 1, 2, … an exceptional collection (E_{0}^{(2j)}, …, E_{m_{j}}^{(2j)}), which is obtained from {E′_{0}, E′_{1}, …, E′_{m}} by mutations, satisfies

inequality ∑_{i=1}^{m_{j}} rkE_{i}^{(2j)} ≥ ∑_{i=1}^{m_{j−1}} rkE_{i}^{(2j−2)}, has m_{j} ≤ m_{j−1}, and represents F by the exact triple

0 ^{}⊕_{i=1}^{m_{j}} x_{i}^{(2j)} E_{i}^{(2j)}  ^{}F ^{}x_{0}^{(2j)} E_{0}^{(2j)}  ^{}0 .  (6-15)

As soon the number m_{j} becomes less than m, we can use the inductive hypothesis. So, we

can assume that m_{j} = m ∀j. Since ∑_{i=0}^{m} rkE_{i}^{(2j)} ≤ rkF by (6-15), we have to get a stabilization:

∑_{i=0}^{m} rkE_{i}^{(2j)} = ∑_{i=0}^{m} rkE_{i}^{(2j−2)}

at some step j. Just to simplify the notations, assume that we got it already for j = 1. So, the inequality (6-13) turns to the equality

rkE_{0}^{(1)} + rkE_{1}^{(1)} = rkE′_{0} + rkE′_{1}.

As we have seen in n^{∘} 6.3.1, this means that x_{0}^{(1)} E_{0}^{(1)} = y_{0} E′_{0}, x_{1}^{(1)} E_{1}^{(1)} = x′_{1} E′_{1}, and (6-11) takes the form

0 ^{}⊕_{i=2}^{m} x′_{i} E′_{i}  ^{}F ^{}y_{0} E′_{0} ⊕ x′_{1} E′_{1}  ^{}0 .

Since ⊕_{i=1}^{m} x′_{i} E′_{i} is extra rigid, Ext_{S_{d}}^{1}(x′_{1} E′_{1}, ⊕_{i=2}^{m} x′_{i} E′_{i}) = 0 and x′_{1} E′_{1} is a direct summand of

F = ⊕_{i=0}^{k} α_{i}F_{i}, that is, x′_{1} = α_{ν} and E′_{1} = F_{ν} for some ν, say for ν = 0. So, we obtain the exact triple

0 ^{}⊕_{i=2}^{m} x′_{i} E′_{i}  ^{}⊕_{i=1}^{k} α_{i}F_{i}  ^{}y_{0} E′_{0}  ^{}0 .

and can apply to it the inductive hypothesis, which says that the exceptional collection

{F_{1}, F_{2}, …, F_{k}} is obtained from {E′_{0}, E′_{1}, …, E′_{m}} and has ∑_{i=1}^{k} rkF_{i} ≤ ∑_{i=2}^{m} rkE′_{i} +rkE′_{0}. Hence,

{F_{0}, F_{1}, …, F_{k}} is obtained from {E′_{0}, E′_{1}, …, E′_{m}} as well and satisfies inequality ∑_{i=0}^{k} rkF_{i} ≤

∑_{i=0}^{m} rkE′_{i} as required. The proposition from n^{∘} 6.2.2 is completely proved.

6.4. Proof for the proposition from n^{∘} 6.2.4. First of all, we need some extra information about the restrictions of exceptional sheaves and exceptional pairs onto exceptional divisors. We collect it in two lemmas n^{∘} 6.4.1, n^{∘} 6.4.2 below. Then, in n^{∘} 6.4.3 we show that any exceptional collection can be obtained from an exceptional collection of locally free sheaves with extra rigid sum. After that we finish the proof in n^{∘} 6.4.6.

6.4.1. LEMMA. Restricting a locally free exceptional sheaf to exceptional curve e_{d} ⊂ S_{d}, we get E|_{e_{d}} = x_{1}� _{e_{d}}(s) ⊕ y_{1}� _{e_{d}}(s + 1), where sx_{1} + (s + 1)y_{1} = c_{1}(E) · e_{d} and x_{1} + y_{1} = rkE.

Proof. Certainly, E|_{e_{d}} = ∑ x_{i}� _{e}(i), because of e_{d} ≅ ℙ_{1}. The exact triple

0 ^{}E^{∗} ⊗ E ⊗� (−e_{d}) ^{}E^{∗} ⊗ E ^{}( E^{∗} ⊗ E) |_{e_{d}} ^{}0

leads to exact sequence of cohomologies:

⋯ ^{}Ext_{S_{d}}^{1}(E, E) ^{}Ext_{S_{d}}^{1}(E|_{e_{d}}, E|_{e_{d}}) ^{}Ext_{S_{d}}^{2}(E, E(−e_{d})) ^{}· · · .

Here Ext_{S_{d}}^{1}(E, E) = 0, because E is exceptional. Since E is μ_{can}-stable and

μ_{can}(E(K_{S_{d}} + e_{d})) = μ_{can}(E) + e_{d} · K_{S_{d}} + K_{S_{d}}^{2} = μ_{can}(E) − 1 + K_{S_{d}}^{2} ,

we have Ext^{2}(E, E(−e_{d})) = Hom^{∗}(E, E(K_{S_{d}} + e_{d})) = 0 as well^{1}. So,

0 = Ext_{S_{d}}^{1}(E|_{e_{d}}, E|_{e_{d}}) = Ext_{e_{d}}^{1}(E|_{e_{d}}, E|_{e_{d}}) = ⊕_{i,j} Ext^{1}(� _{e_{d}}(i),� _{e_{d}}(j))

which implies |i − j| ≤ 1 ∀i, j. □

6.4.2. LEMMA. Let {E, F } be an exceptional pair of locally free sheaves on S_{d} such that

either μ_{can}(F) − K_{S_{d}}^{2} < μ_{can}(E) < μ_{can}(F) or μ_{can}(E) = μ_{can}(F) .

Then E|_{e_{d}} = x_{1}� _{e_{d}}(s) ⊕ x_{2}� _{e_{d}}(s + 1) and F |_{e_{d}} splits either as y_{1}� _{e_{d}}(s) ⊕ y_{2}� _{e_{d}}(s + 1) or as y_{2}� _{e_{d}}(s + 1) ⊕ y_{3}� _{e_{d}}(s + 2); in the last case F (K_{S_{d}})|_{e_{d}} = z_{1}� _{e_{d}}(s) ⊕ z_{2}� _{e_{d}}(s + 1). Proof. The long exact cohomology sequence for the triple

0 ^{}E^{∗} ⊗ F (−e_{d}) ^{}E^{∗} ⊗ F ^{}(E^{∗} ⊗ F)|_{e_{d}} ^{}0

contains: ⋯ ^{}Ext_{S_{d}}^{1}(E, F) ^{}Ext_{e_{d}}^{1}(E_{e_{d}}, F_{e_{d}}) ^{}Ext_{S_{d}}^{2}(E, F (−e_{d})) ^{}· · ·. Here Ext_{S_{d}}^{1}(E, F) = 0 by n^{∘} 5.3.5. Since F and E(K_{S_{d}} + e_{d}) are stable and μ_{can}(E(K_{S_{d}} + e_{d})) = μ_{can}(E) + 1 − K_{S}^{2}, the space Ext^{2}(E, F (−e_{d})) = Hom^{∗}(F, E(K_{S_{d}} + e_{d})) is non zero only if K_{S}^{2} = 1 and F = E(K_{S_{d}} +e_{d}). In this case, the Riemann - Roch formula (see n^{∘} 4.1.1) applied to the equality χ(F, F) = χ(F, E(K_{S_{d}} + e_{d})) = 1 gives rkF = 1, that is,

(E, F) = (� _{S_{d}}(D),� _{S_{d}}(D + K_{S_{d}} + e_{d}))

for some divisor D. Since in this case the pair (F, E) is exceptional as well, we have

(F ⊕ E(K_{S_{d}}))|_{e_{d}} =� _{e_{d}}(e_{d} · D − 1) ⊕� _{e_{d}}(e_{d} · D − 2)

i. e. the lemma holds.

So, we can suppose that Ext_{e_{d}}^{1}(E_{e_{d}}, F_{e_{d}}) = 0. On the other side, the previous lemma predicts the splitting of the kind

E_{e_{d}} = x_{1}� _{e_{d}}(s) ⊕ x_{2}� _{e_{d}}(s + 1) and F_{e_{d}} = y_{1}� _{e_{d}}(s′) ⊕ y_{2}� _{e_{d}}(s′ + 1)

So, either x_{2} = 0, s′ ≥ s−1 or x_{2} ≠ 0, s′ ≥ s. If we apply these estimations to the exceptional pair {F (K_{S_{d}}), E}, for which F (K_{S_{d}})|_{e_{d}} = F_{e_{d}} ⊗� _{e_{d}}(−1) = y_{1}� _{e_{d}}(s′ − 1) ⊕ y_{2}� _{e_{d}}(s′), then we get either y_{2} = 0, s ≥ s′ − 2 or y_{2} ≠ 0 and s ≥ s′ − 1. □

1By n^{∘} 5.2.1, any morphism from E to E(K_{S_{d}} + e_{d}) is either zero or isomorphism (the last is possible only for d = 8, K_{S_{d}}^{2} = 1, but in this case it can be shown that Ext^{2}(E, E(−e_{d})) = 0, comp. with the next proof).

6.4.3. Reduction to the extra rigid locally free collections. First of all, note that

any exceptional collection {G_{1}, G_{2}, …, G_{k}} can be obtained from an exceptional collection of locally free sheaves. Indeed, if all G_{i} are torsion sheaves, then G_{i} = � _{e_{i}}(d_{i}) by n^{∘} 4.3.2. But {� _{e_{1}}(d_{1}), …,� _{e_{r}}(d_{r})} is, clearly, obtained from the exceptional collection

{� _{S_{d}} ( − ∑_{i=1}^{r} d_{i}e_{i} ) , � _{e_{1}}(d_{1}) , …, � _{e_{r}}(d_{r}) } ,

where the first sheaf is locally free. As soon one of G_{i}’s is locally free, we can assume^{1} that this is G_{1} but G_{2} =� _{e_{i}}(d_{i}). The left mutation {L_{G_{1}}G_{2}, G_{1}, G_{3}, …, G_{k}} has one more locally free L_{G_{1}}G_{2} given by exact triple

0 ^{}L_{G_{1}}G_{2}  ^{}Hom(G_{1}, G_{2}) ⊗ G_{1}  ^{}G_{2}  ^{}0 .

Repeating this procedure, we remove all torsion sheaves.

Our next step is to make the collection extra rigid. To this aim, let us fix some extra terminology. We say that an exceptional pair {E, F } is an ext-pair, if Ext^{1}(E, F) ≠ 0. If Ext_{S_{d}}^{>0}(E, F) = 0, we say that {E, F } is a hom-pair. By n^{∘} 5.3.5, the first case is equivalent to inequolity μ_{can}(E) > μ_{can}(F) and the last case — to μ_{can}(E) ≤ μ_{can}(F). An exceptional collection {G_{1}, G_{2}, …, G_{k}} that consists of hom-pairs only is called a hom-collection. This

happens if and only if the sheaf ⊕_{i=1}^{k} G_{i} is extra rigid. For any exceptional collection of locally

free sheaves σ = {G_{1}, G_{2}, …, G_{k}} we put

μ_{0}(σ) =^{def} min_{i} {μ_{can}(G_{i})} , μ_{1}(σ) =^{def} max_{i} {μ_{can}(G_{i})} .

6.4.4. LEMMA. Any exceptional collection on S can be obtained from an appropriate homcollection ζ, of locally free sheaves, such that μ_{1}(ζ) − μ_{0}(ζ) < K_{S_{d}}^{2}. Proof. As we have seen above, it is enough to obtain an arbitrary collection of locally free sheaves σ = {G_{1}, G_{2}, …, G_{k}}.

First of all, using induction on k, let us change σ by a hom-collection σ′ that belongs to the same braid group orbit and has μ_{0}(σ) ≤ μ_{0}(σ′), μ_{1}(σ′) ≤ μ_{0}(σ). For k = 2 an ext-pair (G_{1}, G_{2}) can be replaced just by its left mutation (L_{G_{1}}G_{2}, G_{1}), which is a hom-pair^{2} and satisfies μ_{can}(G_{2}) < μ_{can}(L_{G_{1}}G_{2}) < μ_{can}(G_{1}). In general case, we firstly replace first k − 1 bundles in σ = {G_{1}, G_{2}, …, G_{k}} by a hom-collection τ′ = {G′_{1}, G′_{2}, …, G′_{k−1}} with

μ_{0}(σ) ≤ μ_{0}(τ) ≤ μ_{0}(τ′) and μ_{1}(τ′) ≤ μ_{1}(τ) ≤ μ_{1}(σ) .

If μ_{can}(G′_{k−1}) ≤ μ_{can}(G_{k}) we have done. If μ_{can}(G′_{k−1}) > μ_{can}(G_{k}), then (G′_{k−1}, G_{k}) is an ext-pair and we pass to the left mutation σ_{1} = (G′_{1}, …, L_{G′_{k−1}}G_{k}, G′_{k−1}), which has μ_{can}(L_{G′_{k−1}}G_{k}) < μ_{can}(G′_{k−1}), μ_{0}(σ) ≤ μ_{0}(σ_{1}), and μ_{1}(σ_{1}) ≤ μ_{1}(σ). If this collection is still not a hom-collection, then we take the next left mutation (G′_{1}, …, L_{G′_{k−2}}L_{G′_{k−1}}G_{k}, G′_{k−2}, G′_{k−1}), and so on until we will get a required hom-collection σ′ = {G′_{1}, G′_{2}, …, G′_{k}}, which has μ_{0}(σ′) = μ_{can}(G′_{1}) ≤ ⋯ ≤ μ_{can}(G′_{k}) = μ_{1}(σ′).

1maybe, after a number of the replacements {G_{1}, G_{2}, …, G_{k}} by {G_{2}, …, G_{k}, G_{1}(−K_{S_{d}})}

2Recall that L_{G_{1}}G_{2} is defined as the canonical extension

0  G_{2}  L_{G_{1}}G_{2}  Ext_{S_{d}}^{1}(G_{1}, G_{2}) ⊗ G_{1}  0 .

Now, suppose that μ_{1}(σ′) − μ_{0}(σ′) ≥ K_{S_{d}}^{2} and take a number j such that μ_{can}(G′_{j−1}) < μ_{can}(G′_{j}) = μ_{can}(G′_{j+1}) = ⋯ = μ_{can}(G′_{k}). The exceptional collection

σ′′ = (G_{j}(K_{S_{d}}), …, G_{k}(K_{S_{d}}), G_{1}, …, G_{j−1})

lies in the same braid group orbit and has

μ_{can}(G′_{j}(K_{S_{d}})) = ⋯ = μ_{can}(G′_{k}(K_{S_{d}})) = μ_{can}(G′_{k}) − K_{S_{d}}^{2} ,

i. e. μ_{1}(σ′′)−μ_{0}(σ′′) = μ_{1}(σ′)−μ_{0}(σ′)−K_{S_{d}}^{2}. As above, we can replace σ′′ by a new exceptional hom-collection σ′′′ that has μ_{0}(σ′′′) ≤ μ_{0}(σ′′), μ_{1}(σ′′′) ≤ μ_{1}(σ′′) and so on until we will get the required estimation for the difference μ_{1} − μ_{0}. □

6.4.5. COROLLARY. Any exceptional collection on S can be obtained from an exceptional hom-collection σ = {F_{1}, F_{2}, …, F_{k}}, of locally free sheaves, such that

⊕_{i=1}^{k} F_{i}|_{e_{d}} = x_{0}� _{e_{d}}(s) ⊕ x_{1}� _{e_{d}}(s + 1) .  (6-16)

Proof. Let ζ = {G_{1}, G_{2}, …, G_{k}} be an exceptional collection constructed in the previous proposition. Then each pair {G_{i}, G_{j}} ⊂ ζ satisfies the assumptions of the lemma from n^{∘} 6.4.2. Hence, for some j : 1 ≤ j ≤ k and some s ∈ ℤ, the sheaves G_{ν} are restricted on exceptional divisor e_{d} as

G_{ν}|_{e_{d}} = \{ \begin{matrix}x_{ν}� _{e_{d}}(s) ⊕ y_{ν}� _{e_{d}}(s + 1) & , for ν < j \\ y_{ν}� _{e_{d}}(s + 1) ⊕ z_{ν}� _{e_{d}}(s + 2) & , for ν ≥ j \end{matrix} .

Then the collection σ = {G_{j}(K_{S}), G_{j+1}(K_{S}), …, G_{k}(K_{S}), G_{1}, G_{2}, …, G_{j−1}} is what we need. □

6.4.6. End of proof for the proposition from n^{∘} 6.2.4. To finish the proof, we have to

find, for an exceptional collection σ = {F_{1}, F_{2}, …, F_{k}} obtained in the previous corollary, another exceptional collection {� _{e_{d}}(−1), E_{1}, …, E_{m} } that satisfies the assumptions of the proposition from n^{∘} 6.2.2. To this aim, consider the canonical map

F ^{can}^{}Hom_{S_{d}}(F,� _{e_{d}}(−1))^{∗} ⊗� _{e_{d}}(−1)

Since F |_{e_{d}} = x_{0}� _{e_{d}}(−1) ⊕ x_{1}� _{e_{d}} and Hom_{S_{d}}(F,� _{e_{d}}(−1))^{∗} = Hom_{e_{d}}(F |_{e_{d}},� _{e_{d}}(−1))^{∗}, the restriction of can onto e_{d} is surjective. Hence, can is surjective on S_{d} as well and we have an exact triple

0 ^{}E ^{}F ^{can}^{}x_{0}� _{e_{d}}(−1) ^{}0 ,  (6-17)

which gives the right mutation E = R_{� _{e_{d}}(−1)}F ∈ Ob (^{⊥}� _{e_{d}}(−1) ). Applying Ext( ∗ , F) to (6-17), we get

⋯ ^{}Ext_{S_{d}}^{q}(F, F) ^{}Ext_{S_{d}}^{q}(E, F) ^{}Ext_{S_{d}}^{q+1}(x_{0}� _{e_{d}}(−1), F) ^{}⋯,

§ 7. Helices and Riemann - Roch arithmetics. 53

Since F is extra rigid and

Ext_{S_{d}}^{2}(� _{e_{d}}(−1), F)^{∗} = Ext_{S_{d}}^{0}(F,� _{e_{d}}(−1) ⊗� _{S_{d}}(K_{S_{d}})) = = Ext_{S_{d}}^{0}(F,� _{e_{d}}(−2)) = Ext_{e_{d}}^{0}(F |_{e_{d}},� _{e_{d}}(−2)) = 0

(the last equality follows from (6-16) with s = −1), we have Ext_{S_{d}}^{>0}(E, F) = 0, which implies that E is an extra rigid locally free sheaf (recall that Hom^{•}(E, E) = Hom^{•}(E, F), because L_{� _{e_{d}}(−1)} is the right adjoint functor for the inclusion ^{⊥}� _{e_{d}}(−1)^{⊂} ^{}� _{�oh}^{b} (S_{d})). Thus,

E = ⨁_{i=1}^{m} x_{i} E_{i} for some exceptional collection {E_{1}, E_{2}, …, E_{m}}. This completes the proof. □

§7. Helices and Riemann - Roch arithmetics.

7.1. Arithmetics of K_{0}(ℙ_{n}). Let M = K_{0}(ℙ_{n}) be the Mukai lattice of the projective space and W = M ⊗ ℚ be its complexification. We will write

⟨ E , F ⟩ = ∑(−1)^{ν} dim Ext^{ν}(E, F)

for the Euler scalar product and χ = ( χ_{ij}) = ( ⟨ e_{i} , e_{j} ⟩) for its Gram matrix as soon a base {e_{ν}} is chosen.

7.1.1. Hilbert polynomials model. As a ℤ-module, M can be identified with the module of integer valued polynomials {h ∈ ℚ[t] | h(ℤ) ⊂ ℤ} by taking a class [E] ∈ K_{0}(ℙ_{n}) to its Hilbert polynomial h_{E}(k) = ⟨� (−k) , E ⟩. Under this identification the structure sheaves concentrated on the linear subspaces

{� _{ℙ_{n}},� _{ℙ_{n−1}}, …,� _{ℙ_{1}},� _{ℙ_{0}}}

go to the standard binomial ℤ-basis

{γ_{n}(t), γ_{n−1}(t), …, γ_{0}(t)}

for integer valued polynomials, where we write

γ_{n}(t) =^{def} h_{� _{ℙn}}(t) = \binom{t + n}{n} = \frac{1}{n!}(t + 1)(t + 2) ⋯ (t + n)
· · · · · · · · ·
γ_{k}(t) =^{def} h_{� _{ℙ_{k}}}(t) = \binom{t + k}{k} = \frac{1}{k!}(t + 1)(t + 2) ⋯ (t + k)
· · · · · · · · ·
γ_{1}(t) =^{def} h_{� _{ℙ_{1}}}(t) = \binom{t + 1}{1} = (t + 1)
γ_{0}(t) =^{def} h_{� _{ℙ_{0}}}(t) ≡ 1

Further, the twisting operator T : [E] ⟼ [E(1)] goes to the shift

T = e^{\frac{d}{dt}} : f(t) ⟼ f(t + 1)

and the restriction onto a hyperplane 1 − T^{−1} : [E] ↦ [E|_{ℙ_{n−1}}] goes to the left difference operator

∇ = 1 − e^{− \frac{d}{dt}} : f(t) ⟼ f(t) − f(t − 1)

In particular, the canonical Serre operator from n^{∘} 1.4.3 equals

κ = (−1)^{n}T^{n+1} = (−1)^{n}e^{−(n+1) \frac{d}{dt}}

and is quasiunipotent. This implies (see [Go5]) that M is a lattice of U_{n}-type in the sense of n^{∘} 1.4.4.

7.1.2. Differential operators model. All ℤ-linear maps M ^{}M that commute with the canonical operator form a subalgebra � ⊂ End_{ℤ}(M) that goes to ℤ[∇]/∇^{n+1} under the identification from the previous section. Its rational version � ⊗ ℚ ⊂ End_{ℚ}(W) is nothing but ℚ[D]/D^{n+1}, where D acts on W ⊂ ℚ[t] as \frac{d}{dt}. This leads to an other useful model for M: sending a formal power series A(D) ∈ ℚ[[D]] to A(D)γ_{n}, we identify � = ℤ[∇]/∇^{n+1} with M ⊂ ℚ[t] in such a way that the product of power series goes to the tensor product of sheaves and the involution [E] ⟼ [E^{∗}] corresponds to the sign change D ⟼ −D. In this terms

⟨ A(D) , B(D) ⟩ = A(−D)B(D)γ_{n} (0)

and ⟨ A(−D)h_{1} , h_{2} ⟩ = ⟨ h_{1} , A(D)h_{2} ⟩ ∀ h_{1}, h_{2} ∈ M ⊂ ℚ[t].

7.1.3. COROLLARY. The rational isometry group of K_{0}(ℙ_{n}) ⊗ ℚ ⊂ ℚ[t] consists of differential operators ±e^{A(D)} (mod D^{n+1}), where A(D) ∈ ℚ[[D]] is odd: A(−D) = −A(D). □

7.1.4. Remark. The precise description of the integer isometry group of K_{0}(ℙ_{n}) for general n is still unknown. Some its invariants are described in [Go5]. The conjecture from n^{∘} 1.5.1 asserts that any semiorthonormal collection of n + 1 Hilbert polynomials can be obtained from the simplest one {γ_{n}(t), γ_{n}(t + 1), …, γ_{n}(t + n)} by the isometries and mutations. It is proved only for n = 2, 3. The only known proof for ℙ_{3} (see [No1]–[No3]) is essentially a hard handwork in the integer linear programming style with a lot of quadratic relations.

7.2. Towards Markoff. Now we are going to revisit the discussion started in n^{∘} 1.6 of the introduction. Let N be a free ℤ-module of rank 3 equipped with an unimodular integer bilinear form. In accordance with general results in [Go5], the Jordan normal form of the corresponding

canonical Serre operator κ on V = N ⊗ℂ is either (\begin{matrix}1 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{matrix}), which corresponds to U_{3}-type,

or (\begin{matrix}−1 & 1 & 0 \\ 0 & −1 & 0 \\ 0 & 0 & 1 \end{matrix}), which corresponds to U_{2} ⊕ U_{1} - type, or (\begin{matrix}λ & 0 & 0 \\ 0 & λ^{−1} & 0 \\ 0 & 0 & 1 \end{matrix}), which corresponds

to U_{1} ⊕ W_{2}(λ) - type, where λ ≠ ±1. These cases are separated by tr (κ), which equals 3, −1 and 1 + λ + λ^{−1} respectively. If N admits a semiorthonormal base with the Gram matrix

χ = ( \begin{matrix}1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{matrix} ) .  (7-1)

then tr (κ) = tr (χ^{−1}χ^{t}) = 3 − a^{2} − b^{2} − c^{2} + abc. So, N ⊗ ℂ is an irreducible space of type U_{3} iff the numbers {a, b, c} satisfy the triple Markoff equation:

a^{2} + b^{2} + c^{2} = abc .

All its solutions (see [Ca]) are obtained from the initial solution {3, 3, 3} by changing signs of any two numbers and changing one of three numbers via Vieta theorem: a ↦ bc − a, or b ↦ ac − b, or c ↦ ab − c. To change signs of any two elements in the Gram matrix it is enough to change a sign of one of basic vectors. The mutations

{e_{0}, e_{1}, e_{2}} ⟼ {⟨ e_{0} , e_{1} ⟩ · e_{0} − e_{1} , e_{0} , e_{2}}
{e_{0}, e_{1}, e_{2}} ⟼ {e_{0} , ⟨ e_{1} , e_{2} ⟩ · e_{1} − e_{2} , e_{1}}
{e_{0}, e_{1}, e_{2}} ⟼ {e_{0} , e_{2} , ⟨ e_{1} , e_{2} ⟩ · e_{2} − e_{1}} ,

of a semiorthonormal basis {e_{0}, e_{1}, e_{2}}, change the Gram matrix by the rules

(\begin{matrix}1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{matrix}) ⟼ (\begin{matrix}1 & a & ab − c \\ 0 & 1 & b \\ 0 & 0 & 1 \end{matrix})

(\begin{matrix}1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{matrix}) ⟼ (\begin{matrix}1 & ac − b & a \\ 0 & 1 & c \\ 0 & 0 & 1 \end{matrix})

(\begin{matrix}1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{matrix}) ⟼ (\begin{matrix}1 & b & bc − a \\ 0 & 1 & c \\ 0 & 0 & 1 \end{matrix})

So, any semiorthonormal basis in a Mukai lattice of U_{3}-type can be transformed by the braid group action and changing signs of basic vectors to the semiorthonormal basis with Gram matrix

(\begin{matrix}1 & 3 & 3 \\ 0 & 1 & 3 \\ 0 & 0 & 1 \end{matrix})

which appears in K_{0}(ℙ_{2}) for the basis {� , � (−1), � (1)}. We have established

7.2.1. PROPOSITION. There exists a unique up to an integer isometry Mukai lattice of type U_{3} that admits a semiorthonormal base. This lattice is K_{0}(ℙ_{2}). Up to the signs of basic vectors and integer isometries, the braid group B_{3} acts transitively on the semiorthonormal bases of K_{0}(ℙ_{2}). □

7.2.2. Markoff tree and helices on ℙ_{2}. The description of exceptional vector bundles on ℙ_{2} first obtained in [DLP] can be presented as follows.

First of all, any exceptional vector bundle E is defined up to an isomorphism by its class [E] ∈ M = K_{0}(ℙ_{2}). Indeed, if [E′] = [E], then Hom(E, E′) = Ext^{2}(E, E′) = 0 by the stability but χ(E, E′) = χ(E, E) = 1 > 0 by Riemann - Roch.

Further, 1-dimensional sublattice ⟨E⟩ ⊂ M spanned by an exceptional [E] contains explicitly two vectors e with ⟨ e , e ⟩ = 1, namely [E] and −[E] = [E[1]]. Since any exceptional sheaf E is contained in some exceptional base {E, E_{1}, E_{2}} of � _{�oh}^{b} (ℙ_{2}) (see [DLP], [GoRu]), the 2-dimensional sublattice ⟨^{⊥}E⟩ ⊂ M admits an exceptional base {e_{i}, e_{2}} with e_{ν} = [E_{ν}]. We have seen in n^{∘} 1.7 that all exceptional bases of this lattice are exhausted up to a sign by the mutations^{1} in the pair (E_{1}, E_{2}).

Reducing the Riemann - Roch expansion (4-2) for χ(E_{1}, E_{2}) modulo the orthogonality relations χ(E_{1}, E) = χ(E_{2}, E) = 0 we get immediately χ(E_{1}, E_{2}) = 3rk(E). In other words, the restriction of χ onto ⟨^{⊥}E⟩ depends only on rkE, and ranks r_{0}, r_{1}, r_{2} in an exceptional triple {E_{0}, E_{1}, E_{2}} satisfy the ordinary (non triple) Markoff equation

r_{0}^{2} + r_{1}^{2} + r_{2}^{2} = 3r_{0}r_{1}r_{2}  (7-2)

Moreover, it follows easily from n^{∘} 1.7.2 that if rkE ≥ 2, then there exists a unique semiorthogonal base {E_{1}, E_{2}} in ⟨^{⊥}E⟩ such that both rkE_{1}, rkE_{2} < rkE. Hence, applying the mutations that strictly decrease the rank of a muted bundle, we transform in a unique way any exceptional triple to an exceptional triple of invertible sheafs, that is, to {� (−1), � , � (1)} modulo a twist.

In other words, starting with the initial exceptional triple {� (−1), � , � (1)} and applying only the mutations

\begin{matrix}\{E_{0}, E_{1}, E_{2}\} \\ \{E_{1}, R_{E_{1}}E_{0}, E_{2}\} & \frac{}{g_{1}} & \{E_{0}, L_{E_{1}}E_{2}, E_{1}\} & ^{^{g_{2}^{−1}}} \end{matrix}

we get a binary tree whose vertices are labelled by exceptional triples {E_{0}, E_{1}, E_{2}} in such a way that each twist class of exceptional triples is represented exactly by one vertex. This three has a symmetry w. r. t. the vertical axis. It takes {E_{0}, E_{1}, E_{2}} to {E_{2}^{∗}, E_{1}^{∗}, E_{0}^{∗}}. So, taking the right half of this three we get a three that classifies exceptional triples up the twists and the duality. It is called the Markoff three, because it classifies all positive solutions (r_{0}, r_{1}, r_{2}) of the Markoff equation (7-2) such that r_{1} ≥ r_{0}, r_{2}.

7.2.3. Markoff’s problem and A. N. Tyurin’s conjecture. A famous problem in the

geometry of numbers is: are there two different solutions (r_{0}, r, r_{2}), (r′_{0}, r, r′_{2}) (with the same middle term r) of the Markoff equation (7-2) such that r ≥ r_{0}, r_{2}, r′_{0}, r′_{2}? This problem has more than 100 years history with a lot of incorrect proofs like for the great Fermat theorem. The positive answer (conjectured by Markoff) could give a far improvements in the Diophantine approximations. Our previous discussion shows that the Markoff problem is equivalent to the following question asked by A. N. Tyurin in 1980’th before the helix theory was developed: is an exceptional vector bundle on ℙ_{2} uniquely up to twists and duality defined by its rank? Indeed, careful control over the slopes shows that any exceptional vector bundle E such that μ(E) = c_{1}(E)/rk(E) lies in [0, 1/2] presented as a middle element of exceptional triple at precisely one vertex of the Markoff three. So, the Tyurin’s question is: can the same r appear as the middle term in two different Markoff triples (r_{0}, r, r_{2}) ordered in such a way that the middle term is maximal? The Markoff problem is still open …

1in this case mutations and isometries essentially coincide

7.3. Helices and local systems. In this section we discuss an interaction between helices and rigid local systems over ℙ_{1} \ {some points} discovered and studied by V. Golyshev in a very recent paper [Gl]. We follow below his notations and terminology.

7.3.1. Rigid local systems. Let U ^{⊂} ^{j}^{}ℙ_{1} be an (analytic) open set with a finite complement ℙ_{1} \U = {x_{0}, x_{1}, …, x_{n}}. We fix n+1 basic loops {γ_{0}, γ_{1}, …, γ_{n}} ⊂ π_{1}(U) coming from some base point in U and passing around the points x_{ν}. A rank r local system � over U is a locally trivial vector bundle of rank r over U equipped with a flat GL_{r}(ℂ)-connection, or equivalently, an isomorphism class of representations π_{1}(U) ^{φ}^{}GL_{r}(ℂ) given by the monodromy of� . In other words, a local system is defined by n + 1 linear operators φ(γ_{ν}) ∈ GL_{r}(ℂ) such that φ(γ_{0})φ(γ_{1}) ⋯ φ(γ_{n}) = 1. A local system given by a representation φ is called rigid, if any other representation φ′ such that

∃ g_{ν} ∈ GL_{r}(ℂ) : φ(γ_{ν}) = g_{ν}φ(γ_{ν})g_{ν}^{−1} ∀ ν

is isomorphic to φ, i. e. ∃ g ∈ GL_{r}(ℂ) : φ(γ) = gφ(γ)g^{−1} ∀ γ ∈ π_{1}(U)).

An irreducible local system� of rank r ≥ 2 is rigid iff χ ( ℙ_{1}, j_{∗}(� ^{∗} ⊗� ) ) = 2. Moreover, rigidness is a local property: it is sufficient to check it only for φ′ closed enough to φ (see [Kt, sec. 1.1] for all the details).

A rigid irreducible local system φ such that all φ(γ_{ν}) are quasiuniptent^{1} operators is called a Katz local system. A pull back � ′ = p^{∗}� of a Katz local system � along a covering U′ ^{p} [figure] U is called a quasi Katz local system. Such the local systems was studied by N. Katz in [Kt], where among other things any such a system was realized geometrically in the fiber cohomologies of appropriate pencil of algebraic varieties.

7.3.2. LEMMA. Any two complex coprime polynomials of the same degree with unit constant terms: p(t) = 1 + p_{1}t + ⋯ + p_{r+1}t^{r}, q(t) = 1 + q_{1}t + ⋯ + q_{r+1}t^{r} define a unique irreducible rigid local system φ of rank r = deg p = deg q on ℙ_{1}{ 0 , 1 , ∞ } such that its local monodromies T = φ(γ_{0}), S = φ(γ_{1}) satisfy the conditions

(a) S is a quasireflection, i. e. rk(1 − S) = 1
(b) a tilt vector of S is cyclic for T,
i. e. im (1 − S), T (im (1 − S)), …, T^{r−1}(im (1 − S)) span ℂ^{r}
(c) det(1 − t T) = q(t)
(d) det(1 − t ST) = p(t)

Proof. Let us fix a base {v_{1}, v_{2}, …, v_{r}} in V = ℂ^{r} that, by the definition, is cyclic for T with the cyclic vector v_{1}, i. e. v_{ν} = T^{ν−1}v_{1} for 1 ≤ ν ≤ r. Further, define ξ ∈ V^{∗} by the prescription ξ(v_{ν}) = h_{ν}, where h_{ν} are the starting coefficients of a formal power series

H(t) = 1 + ∑_{ν≥1} h_{ν}t^{ν} =^{def} \frac{p(t)}{q(t)}

1that is, have a form ζ + η, where η is nilpotent and ζ is diagonalizable and satisfies ζ^{m} = 1 for some integer m

Then S =^{def} 1 − ξ ⊗ v_{1} and T satisfy the conditions (a)–(d), and rank 1 operator 1 − S = ξ ⊗ v is uniquely recovered from p and q, because of obvious determinantal identity

1 + ∑_{ν≥1} ξ(v)t^{ν} = \frac{det(1 − tT S)}{det(1 − tT)}

This proves the existence of S, T, T S, and their uniqueness up to a simultaneous conjugation. If W ⊂ V is invariant subspace, then S|_{W} = Id_{W}, because of otherwise (1 − S)W ≠ 0 and W ⊃ ∑ T^{ν}(1 − S)W = V. So, T |_{W} = ST |_{W} have common eigenvalue and their characteristic polynomials can not be coprime. □

7.3.3. Mutations and transvections. Let M be a Mukai lattice, ⟨ ∗ , ∗ ⟩ be its unimodular bilinear form, and ε = ±1. We put

(v, w)_{ε} =^{def} ⟨ v , w ⟩ + ε ⟨ w , v ⟩ ,

which is symmetric for ε = 1 and skew symmetric for ε = −1. For any e ∈ M such that ⟨ e , e ⟩ = 1 a map

M ^{σ_{e}} ^{}M : σ_{e}^{ε}(v) = v − (e, v)_{ε}e

is called a transvection w. r. t. e. This is, certainly, a quasireflection. We clearly have

σ_{e}^{ε}(v) = −L_{e}(v) − ε ⟨ v , e ⟩ e  (7-3)

where L_{e}(v) = ⟨ e , v ⟩ e − v is the left mutation w. r. t. e (see n^{∘} 1.4.1). In particular, if {e_{1}, e_{2}} ⊂ M is an exceptional pair, then σ_{e_{1}}^{ε}(e_{2}) = −L_{e_{1}}(e_{2}).

7.3.4. LEMMA. For any exceptional base {e_{0}, e_{1}, …, e_{n}} ⊂ M

σ_{e_{0}}^{ε}σ_{e_{1}}^{ε} ⋯ σ_{e_{n}}^{ε} = (−1)^{n+1}εκ

where M ^{κ}^{}M is the canonical Serre operator (see n^{∘} 1.4.3). Proof. Consequently applying (7-3) and using (1-3) and (1-5), we get^{1}:

σ_{e_{0}}^{ε}σ_{e_{1}}^{ε} ⋯ σ_{e_{n}}^{ε}(v) =

= (−1)^{n+1}L_{e_{0}}L_{e_{1}} ⋯ L_{e_{n}}(v) − ε ∑_{ν}(−1)^{ν} ⟨ v , e_{n−ν} ⟩ L_{e_{0}}L_{e_{1}} ⋯ L_{e_{n−ν−1}}e_{n−ν} =

= −ε ∑_{ν}(−1)^{ν} ⟨ v , e_{n−ν} ⟩ e_{ν}^{∨} = (−1)^{n+1}εκ(v) ,

because of L_{e_{0}}L_{e_{1}} ⋯ L_{e_{n}}(v) ∈ ⟨ e_{0}, e_{1}, …, e_{n} ⟩^{⊥} = 0. □

7.3.5. Quasi Katz local systems associated with helices. Let M be a Mukai lattice of

type U_{n+1} (for example, M = K_{0}(ℙ_{n})). Then (see [Go5]) the canonical operator on M has a form κ = (−1)^{n}Id+η, where and η is maximal nilpotent, that is, η^{n+1} = 0 but im η^{n} = w_{0} ≠ 0. Let us put ε = (−1)^{n+1}. Then the (anti) symmetric bilinear form on W = M ⊗ ℂ considered in the previous section, i. e.

(v, w)_{ε} = ⟨ v , w ⟩ + ε ⟨ w , v ⟩ = (−1)^{n} ⟨ v , ((−1)^{n}Id − κ)w ⟩ ,

1comp. with n^{∘} 1.4.3 and n^{∘} 2.8.1

has 1-dimensional kernel ⟨w_{0}⟩, spanned by w_{0}. We put V =^{def} W/⟨w_{0}⟩, write (∗, ∗) for the non degenerate bilinear form induced on V by (∗, ∗)_{ε} and write σ_{e}(v) = v − (e, v) e (mod w_{0}) for the corresponding transvections.

For any exceptional base \overline{e} = \{e_{0}, e_{1}, …, e_{n}\} ⊂ M let � _{M,\overline{e}} be a local system on U = ℙ_{1} \ { 1 ζ, …, ζ^{n}, ∞}, where ζ is a complex primitive (n + 1)-th root of unity, associated with a representation σ : π_{1}(U) ^{}GL(V) given by the prescription

\begin{matrix}σ(γ_{ζ^{ν}}) & =^{def} σ_{e_{ν}} : v ⟼ v − (e_{ν}, v) · e_{ν} for 0 ≤ ν ≤ n \\ σ(γ_{∞}) & =^{def} (σ_{e_{0}}σ_{e_{1}} ⋯ σ_{e_{n}})^{−1} \end{matrix} (mod w_{0})

7.3.6. PROPOSITION. The local system � _{M,\overline{e}} constructed by the above rules for M = K_{0}(ℙ_{n}) ⊗ ℂ and \overline{e} = \{� , � (1), …, � (n)\} is quasi Katz. Proof. Consider a local system� on U′ = ℙ_{1} \ { 0 , 1 , ∞ } whose local monodromy around 0 is induced by twisting operator, which takes E to E(1), and local monodromy around 1 is the transvection σ_{e_{0}} w. r. t. to the class [� ]. In other words, � is associated with the representation ϱ : π_{1}(U′) ^{}GL(W) that sends

\begin{matrix}ϱ(γ_{0}) & =^{def} T : [E] ⟼ [E ⊗� (1)] \\ ϱ(γ_{1}) & =^{def} σ_{e_{0}} : v ⟼ v − (e_{0}, v) · e_{0} \end{matrix} (mod w_{0})

Note that T and S = σ_{e_{0}} satisfy the conditions of lemma from n^{∘} 7.3.2, that is, � is rigid and irreducible. Now consider (n + 1)-th degree cover

ℙ_{1} \ {0, 1 ζ, …, ζ^{n−1}, ∞} ^{p} [figure] ℙ_{1} \ { 0 , 1 , ∞ }

given by z ⟼ z^{n+1} in the standard affine chart near {0} ∈ ℙ_{1}. Then p^{∗}� can be continued to the origin {0} ⊂ ℙ_{1}, because all the eigenvalues of κ = ϱ(γ_{0}) are some (n + 1)-th roots of unity. Let us write � ′ for this continuation. Actually, � ′ coincides with � _{M,\overline{e}}. Indeed, the monodromy around the infinity in the both cases is given by the canonical operator T^{n+1} and the monodromies around the roots of unity are S, T ST^{−1}, T^{2}ST^{−2}, …, T^{n}ST^{−n}, because of σ_{e_{ν}} = σ_{T^{ν}e_{0}} = T^{ν}σ_{e_{0}}T^{−ν} □

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