Birationally rigid Fano complete intersections

A.V.Pukhlikov

Steklov Mathematics Institute Gubkina, 8 117966 Moscow RUSSIA e-mail: pukh@mi.ras.ru and Max-Planck-Insitut für Mathematik Vivatsgasse 7 53111 Bonn GERMANY

January 3, 2000

Abstract

We prove that general Fano complete intersections V = V_{d_{1}·...·d_{k}} = F_{1} ∩ … ∩ F_{k} ⊂ P^{M+k}, where F_{i} ⊂ P^{M+k} is a hypersurface of degree d_{i} ≥ 2 and the integers d_{1}, …, d_{k} satisfy the relation d_{1} + … + d_{k} = M + k (so that V is of index 1) are birationally superrigid for M ≥ 2k + 1, k ≥ 2. In particular, they cannot be fibered by a rational map into uniruled varieties (over a positive-dimensional base), each birational map χ: V − − → V′ onto a Fano variety V′ with Q-factorial terminal singularities is biregular in codimension one and the groups of birational and biregular self-maps coincide: Bir V = Aut V. The proof is based upon the method of maximal singularities and the techniques of hypertangent systems.

Introduction

0.1. Recall that a smooth Fano variety X of dimension ≥ 3 with Pic X = ZK_{X} is said to be birationally superrigid, if for each birational map χ: X − − → X′ onto a variety X′ of the same dimension, smooth in codimension one, and each linear system Σ′ on X′, free in codimension 1 (that is, codim Bs Σ′ ≥ 2), the inequality

c(Σ, X) ≤ c(Σ′, X′)

holds, where Σ = (χ^{−1})_{∗}Σ′ is the proper inverse image of Σ′ on X with respect to χ, and c(Σ, X) = c(D, X) stands for the threshold of canonical adjunction

c(D, X) = sup{b/a|b, a ∈ Z_{+} \ {0}, |aD + bK_{X}| ≠ ∅}

D ∈ Σ, and similarly for Σ′, X′.

If X is birationally superrigid, then for each Fano variety X′ of the same dimension with Q-factorial terminal singularities each birational map χ: X − − → X′ (provided there are such maps) is biregular in codimension 1. Moreover, if Pic X′ ⊗ Q = QK_{X′}, then χ, χ^{−1} are isomorphisms in codimension 1. Furhermore, there exists no rational map ψ: X − − → S onto a positive dimensional algebraic variety S, dim S < dim X, a generic fiber of which, ψ^{−1}(s), s ∈ S, is a uniruled variety. In particular, the variety X is non-rational. Finally, the groups of birational and biregular self-maps coincide: Bir V = Aut V. These implications of birational rigidity are well known; see, for instance, [IM,IP,P2,CPR,…].

In the present paper we consider Fano varieties of the type

V =  V_{d_{1}·d_{2}·...·d_{k}}  ⊂ P^{M+k}
∥
F_{1} ∩ F_{2} ∩ … ∩ F_{k},

where F_{i} ⊂ P^{M+k} are hypersurfaces of degree d_{i} ≥ 2. The integers d_{i} satisfy the relation d_{1} + … + d_{k} = M + k, k ≥ 2, and V is assumed to be smooth. In other words, V is a Fano complete intersection of index 1. The ground field is assumed to be the field of complex numbers C. Our main result is the following

Theorem. For M ≥ 2k + 1 a generic variety V is birationally superrigid.

Generic Fano hypersurfaces of index 1 (which correspond to the value k = 1 in the notations above, i.e. the hypersurfaces V_{M+1} ⊂ P^{M+1}) were proved to be birationally superrigid in [P2] (for M = 3 it is the classical theorem of V.A.Iskovskikh and Yu.I.Manin on the three-dimensional quartic [IM], for M = 4 see [P1]).

For convenience we assume that d_{1} ≤ d_{2} ≤ … ≤ d_{k}. The condition M ≥ 2k + 1 implies the inequality d_{k} ≥ 4. The dimension of the variety V is equal to M ≤ 5. For M = 5 there are two possible cases: V_{2·5} and V_{3·4}.

In dimension three there are two families of complete intersections: V_{2·3} ⊂ P^{5} and V_{2·2·2} ⊂ P^{6}. Birational rigidity of a general variety in the first family was proved in [IP], and its group of birational self-maps Bir V_{2·3} ≠ Aut V_{2·3} was described in [I,IP]. The variety V_{2·2·2} has a lot of conic bundle structures, and geometry of its birational correspondences has not yet been studied.

Conjecture. A smooth Fano complete intersection of index 1 and dimension ≥ 4 in the projective space is birationally superrigid.

There is an immense variety of families of Fano complete intersections with Qfactorial terminal singularities in the weighted projective spaces [Fl]. Birational superrigidity of general varieties of the type

V_{m·2l} ⊂ P(\underbrace{1, …, 1,}_{M+2} l),

m + l = M + 1, was proved in [P3] (these varieties are smooth). It seems that the Conjecture is true for the weighted complete intersections as well.

0.2. The theorem is proved in the following way. By the method of maximal singularities [IM,IP,P2,P3] we reduce the problem of whether a given variety is (super)rigid to behaviour of algebraic cycles on this variety. The following proposition gives a convenient sufficient condition of birational superrigidity.

Proposition 1. Let X be a smooth Fano variety with Pic X = ZK_{X}. Assume that for each point x ∈ X, each irreducible cycle Y ⊂ X of codimension two and each linear system Σ on X without fixed components the following two properties are satisfied:

(i) mult_{Y} Σ ≤ n;

(ii) mult_{x} Y ≤ (4/ deg X) deg Y, where n ∈ Z_{+} is defined by the inclusion Σ ⊂ | − nK_{X}|, deg X = (−K_{X})^{dim X} and deg Y = (Y · (−K_{X})^{dim Y}).

Then the variety X is birationally superrigid.

For the proof, see [P2]. In that paper the proof is carried out by means of the method of maximal singularities, which goes back to the classical paper of V.A.Iskovskikh and Yu.I.Manin [IM]. Another proof, based upon Shokurov’s connectedness theorem and inversion of adjunction [Kol] can be extracted from [C].

We prove the main theorem of the present paper, checking the conditions (i) and (ii) for a general complete intersection V = V_{d_{1}·...·d_{k}}. The property (i) obviously follows from the Lefschetz theorem, since dim V = M ≥ 5. The property (ii) is proved by means of the method of hypertangent systems, which was introduced in [P2]. This is the principal part of our arguments, making the contents of Sec. 1.

In order to apply the method of hypertangent systems, we need the complete intersection V to satisfy a certain regularity condition (Sec. 1.1), which is analogous to the regularity condition of the papers [P2,P3]. Therefore it is necessary to verify that a general complete intersection satisfies the regularity condition at each point. This is done in Sec. 2, thus making the proof of our theorem complete.

Remark. Cheltsov has noted [Ch], that the method of maximal singularities gives not only a description of all the structures of Fano fibrations on a given variety, but also a description of all the structures of K-trivial fibrations. In particular, since a general complete intersection V ⊂ P^{M+k} with M ≥ 2k + 1 satisfies the property (ii), K–trivial fibrations on V are determined by linear systems Σ ⊂ | − nK_{V} |, which either have a base cycle Y ⊂ V of codimension 2 such that mult_{Y} Σ = n, or have a base cycle Y of codimension 3 such that mult_{Y} Σ = 2n. The second case can be easily excluded by the arguments of Sec. 1. In the first case, it is easy to see that Y is numerically equivalent to a section of V by a plane of codimension 2 in P, whereas the system Σ defines a rational map φ_{Σ}: V − − → P^{1}. In other words, each structure of a K-trivial fibration on V has one-dimensional base P^{1}. It seems that all these structures are just pencils of hyperplane sections of V. For the Fano hypersurfaces V_{M+1} ⊂ P^{M+1} this is true [Ch] and the proof is obtained by the methods of [P2]. For the complete intersections, the techniques of [P2] should be improved.

0.3. Acknowledgements

I started to work with Fano complete intersections in 1997 during my stay at Max-Planck-Institut für Mathematik. In [P2] it was pointed out that the techniques developed for the Fano hypersurfaces should work for complete intersections, too. However, it was not until now that all the details have been clarified. The author thanks the Institute for the excellent conditions of work.

The present research was partially supported by the RBRF grants 99-15-96013, 99-01-01132, 99-15-96146 and the INTAS-OPEN grant 97/2072.

1 Proof of the theorem

1.1 The regularity condition

Fix integers k ≥ 2, M ≥ 2k + 1, d_{k} ≥ … ≥ d_{1} ≥ 2, where d_{1} + … + d_{k} = M + k. The sympol P stands for the projective space P^{M+k} over the field C of complex numbers. Let

f_{i} ∈ H^{0}(P, O_{P}(d_{i})) \ {0}, i = 1, …, k,

be homogeneous polynomials on P. The symbol V (f_{1}, …, f_{k}) = V (f_{∗}) denotes the closed algebraic set {f_{1} = … = f_{k} = 0}. Let C^{M+k} ⊂ P be an open affine set, equipped with a system of coordinates (z_{1}, …, z_{M+k}) with the origin at the point x ∈ P. Set

f_{i} = q_{i,0} + … + q_{i,d_{i}},

where q_{i,a} are homogeneous of degree a in (z_{∗}). Obviously, x ∈ V (f_{∗}) if and only if q_{1,0} = … = q_{d_{k},0} = 0.

Definition 1. A point x ∈ V (f_{1}, …, f_{k}) is called a regular point of the complete intersection V (f_{∗}), if the set of polynomials

{q_{i,j}|1 ≤ i ≤ k, 1 ≤ j ≤ d_{i}, (i, j) ≠ (k, d_{k})},

which consists of all the polynomials q_{i,j}, but the last one q_{k,d_{k}}, makes a regular sequence in O_{x,C^{M+k}} = O_{x,P}.

In other words, the system of equations

{q_{i,j} = 0|(i, j) ≠ (k, d_{k})}

defines a curve in C^{M+k}. Since the polynomials q_{i,j} are homogeneous, this curve is a finite union of lines in C^{M+k}, passing through the origin. In particular, on V (f_{∗}) there are at most a finite number of lines passing through a regular point.

Set

H = ∏_{i=1}^{k} (H^{0}(P, O_{P}(d_{i})) \ {0})

to be the set of sequences (f_{1}, …, f_{k}) of non-zero homogeneous polynomials of degrees d_{1}, …, d_{k}, respectively.

Proposition 2. There exists a non-empty Zariski open set U_{reg} ⊂ H such that for (f_{1}, …, f_{k}) ∈ U_{reg} the complete intersection

V = V (f_{1}, …, f_{k}) ⊂ P

is regular at each point.

The proof is given in Sec. 2.

We will prove our theorem in the following form: for (f_{∗}) ∈ U_{reg} the complete intersection V = V (f_{∗}) is birationally superrigid.

Let us fix the complete intersection V and check that the conditions (i) and (ii) of Proposition 1 are satisfied for V. This is enough for the theorem.

Let Σ ⊂ | − nK_{V} | be a linear system of divisors without fixed components, Y ⊂ Bs Σ its base cycle of codimension 2. Take two general divisors, D_{1}, D_{2} ∈ Σ and consider the algebraic cycle of their scheme-theoretic intersection:

Z = (D_{1} • D_{2}) = aY + ∑ a_{i}Y_{i},

where a_{i} ≥ 1, Y_{i} ≠ Y are some irreducible cycles of codimension 2. Obviously, a ≥ (mult_{Y} Σ)^{2}. Compute the degrees:

n^{2} deg V = deg Z = a deg Y + ∑ a_{i} deg Y_{i},

where deg V = d_{1} … d_{k}. By the Lefschetz theorem Y is numerically equivalent to mK_{V}^{2} , m ≥ 1, so that deg Y = m deg V, whence

(mult_{Y} Σ)^{2} ≤ a ≤ n^{2}/m ≤ n^{2},

which is what we need. Q.E.D. for the condition (i) of Proposition 1.

1.2 Hypertangent linear systems

Let us prove that for a regular point x ∈ V and an irreducible cycle C ⊂ V of codimension 2 the following estimate is true:

\frac{mult_{x}}{deg} C ≤ \frac{4}{deg V} .  (1)

Remark. The estimate (1) is exact: just take C = H_{1} ∩ H_{2} ∩ V, where H_{i} are distinct tangent hyperplanes to V at the point x. We get: deg C = deg V, mult_{x} C = 4.

Proof of the inequality (1). Fix an open affine set A ≅ C^{M+k} ⊂ P and a system of coordinates (z_{∗}) with the origin at the point x on A. Set

f_{i,α} = q_{i,1} + … + q_{i,α}.

The symbol s_{a}, a ∈ Z_{+}, stands for an arbitrary homogeneous polynomial of degree a in (z_{∗}). Consider the linear systems

Λ_{i,j}^{A} = | ∑_{α=1}^{min{j,d_{i}−1}} f_{i,α}s_{j−α}|

on A. Taking the closure of a general divisor of the system Λ_{i,j}^{A} in P, we get the linear system Λ_{i,j}^{P} on P. Obviously,

Bs Λ_{i,j}^{A} = {q_{i,α} = 0|1 ≤ α ≤ min{j, d_{i} − 1}},
Bs Λ_{i,j}^{P} = \overline{Bs Λ_{i,j}^{A}}.

Set

Λ_{j}^{P} = ∑_{i=1}^{k} Λ_{i,j}^{P}.

For this linear system we get

Bs Λ_{j}^{P} = \overline{\{q_{i,α} = 0|1 ≤ α ≤ min\{j, d_{i} − 1\}\}}

and

c(j) = codim_{P} Bs Λ_{j}^{P} = ∑_{i=1}^{k} min{j, d_{i} − 1}.  (2)

We get Bs Λ_{d_{k}−1}^{P} = Bs Λ_{d_{k}}^{P} = … = Bs_{∞}^{P} and codim_{P} Bs_{∞}^{P} = M.

Definition 2. The linear system

Λ_{j} = Λ_{j}^{P}|_{V}

is called the j-th hypertangent system at the point x.

Now we must estimate the codimension of the base set of hypertangent systems. A priori one can assert only that codim_{V} Bs Λ_{j} ≥ codim_{P} Bs Λ_{j}^{P} − k: when a system is restricted from P to V, the codimension of the base set may decrease. However, in our case the regularity condition improves the matter.

Lemma 1. (i) codim_{V} Bs Λ_{1} = codim_{P} Bs Λ_{1}^{P}.

(ii) For j ∈ {2, …, d_{k} − 2}

codim_{V} Bs Λ_{j} = codim_{P} Bs Λ_{j}^{P}  (3)

in a neighborhood of the point x.

(iii) For each j we have the estimate

codim_{V} Bs Λ_{j} ≥ codim_{P} Bs Λ_{j}^{P} − 1.

Proof. It is easy to see that for α ≤ d_{i} − 1

f_{i,α}|_{V ∩A} = (q_{i,α+1} + … + q_{i,d_{i}})|_{V ∩A}.  (4)

Set T = T_{x}V ≅ C^{M}. Now we introduce the auxiliary linear systems

Λ_{i,j}^{T} = | ∑_{α=1}^{min{j,d_{i}−1}} q_{i,α+1}s_{j−α}|_{T} ,

Λ_{j}^{T} = ∑_{i=1}^{k} Λ_{i,j}^{T}.

These are linear systems of cones of degree j + 1 on the tangent space T. By the regularity condition and (4)

Λ_{j}^{T} = T_{x}Λ_{j}.

See Sec. 2.3 for the definitions and (elementary) facts of the theory of tangent cones to divisors, cycles and linear systems. (Here we need the regularity condition to conclude that the restriction q_{i,α+1}|_{T} is a non-zero polynomial.) Obviously,

Bs Λ_{j}^{T} = {q_{i,α+1}|_{T} = 0, 1 ≤ α ≤ min{j, d_{i} − 1}} ⊂ T.

Again by the regularity condition we get

codim_{T} Bs Λ_{j}^{T} = codim_{P} Bs Λ_{j}^{P}

for j ∈ {1, …, d_{k} −2}. The case j = d_{k} −1 is excluded because q_{k,d_{k}} is excluded from the regularity condition. Applying Lemma 5, we get both (ii) and the local version of (i): so far we can assert only that the equality codim_{V} Bs Λ_{1} = codim_{P} Bs Λ_{1}^{P} = k is true in a neighborhood of the point x.

The tangent hyperplane to the hypersurface f_{i} = 0 at the point x is given by the equation q_{i,1} = 0. Denote it by T_{i}. Obviously,

Bs Λ_{1} = T_{1} ∩ … ∩ T_{k} ∩ V ∋ x,

where, as we have already established, this cycle is of codimension k in a neighborhood of the point x. However, it is easy to see that for each i, 1 ≤ i ≤ k, the cycle T_{1} ∩ … ∩ T_{i} ∩ V is irreducible by the Lefschetz theorem. Eventually we get that Bs Λ_{1} is an irreducible cycle of codimension k. The part (i) is completely proved.

Let us prove (iii). The base set of the system Λ_{j}^{P} is given by a system of c(j) equations and has codimension c(j), see (2). The base set of the system Λ_{j} is given by the same equations, restricted to V, and c(j) ≤ dim V. This implies that codim_{V} Bs Λ_{j} − codim_{P} Bs Λ_{j}^{P} is a non-increasing function of the parameter j. (For a ≥ b the codimension codim_{V} Bs Λ_{a} is higher than codim_{V} Bs Λ_{b} by, at most, the number of the new equations, that is, c(a) − c(b).) Thus it is sufficient to prove (iii) for j = ∞: dim(Bs_{∞}^{P})|_{V} ≤ 1.

Indeed, the base subscheme (Bs_{∞}^{P})|_{V} is given on V by the system of equations

{q_{i,α} = 0|1 ≤ α ≤ d_{i} − 1}.

Therefore, as a subscheme of the projective space P (Bs_{∞}^{P})|_{V} is given by the system

{q_{i,α} = 0|1 ≤ α ≤ d_{i}}.

of M + k equations. By the regularity condition (Bs_{∞}^{P})|_{V} ≤ 1, which is what we need. Q.E.D. for Lemma 1.

Let us come back to the proof of inequality (1). For j ≥ 1 set w^{+}(j) = ♯{i, 1 ≤ i ≤ k | j ≤ d_{i} − 1}. Furthermore, set w(j) = w^{+}(j) − 1 for j = 1, 2 and d_{k} − 1, and w(j) = w^{+}(j) for the other values of j. Obviously, w^{+}(1) = k and

c(j) = ∑_{α=1}^{j} w^{+}(α)

for each j ≥ 1. For 1 ≤ j ≤ d_{k} − 1 we introduce the spaces

Δ_{j} = \underbrace{Λ_{j} × … × Λ_{j}}_{w(j)},

parametrizing the sets (D_{j,1}, …, D_{j,w(j)}) ∈ Δ_{j} of divisors from the linear systems Λ_{j}. Set

Δ = ∏_{j=2}^{d_{k}−1} Δ_{j}.

1.3 Degrees and multiplicities

Let R ⊂ V be an irreducible cycle of codimension k + 1.

Lemma 2. For a generic set (D_{∗,∗}) ∈ Δ the set-theoretic intersection

R^{∗} = R ⋂ ( ⋂_{j=2}^{d_{k}−1} ⋂_{a=1}^{w(j)} D_{j,a})

is one-dimensional in a neighborhood of the point x.

Proof. First of all, note that the set (D_{∗,∗}) consists of w(2) + … + w(d_{k} − 1) = M − k − 2 divisors, so that dim R^{∗} ≥ 1. For b ∈ {2, …, d_{k} − 1} set

R_{b} = R ⋂ ( ⋂_{j=2}^{b} ⋂_{a=1}^{w(j)} D_{j,a}) .

Let us prove by (ascending) induction on b that for a general set (D_{∗,∗}) ∈ ∏_{j=2}^{b} Δ_{j} we have

codim R_{b} = 2 + ∑_{j=2}^{b} w(j)  (5)

in a neighborhood of the point x. In what follows we omit the words “in a neighborhood of the point x” and simply speak about codimension. However, during the proof of Lemma 2 we always mean the codimension near the point x.

Set for e ∈ {0, 1, …, w(b)}

R_{b,e} = R ⋂ ( ⋂_{j=2}^{b−1} ⋂_{a=1}^{w(j)} D_{j,a}) ⋂ ( ⋂_{a=1}^{e} D_{b,a}) .

In particular, R_{b,w(b)} = R_{b+1,0} = R_{b}. Let us prove by induction on e, that for a generic collection (D_{∗,∗}) the following equality

codim R_{b,e} = k + 1 + ∑_{j=2}^{b−1} w(j) + e  (6)

holds. For b = 2, e = 0 (6) is true by assumption. Take e ≤ w(b) − 1, and assume that (6) is true. By (3),

codim_{V} Bs Λ_{b} = c(b) = ∑_{j=1}^{b} w^{+}(j) = k + 1 + ∑_{j=2}^{b−1} w(j) + w(b),

so that a generic divisor D_{b,e+1} ∈ Λ_{b} contains none of the components of the set R_{b,e}, which contain the point x. This proves (6).

The equality (6) implies (5) automatically. Q.E.D. for the lemma.

Fix an arbitrary collection (D_{∗,∗}) ∈ Δ, satisfying the claim of Lemma 2. Let R_{b} and R_{b,e} mean the same as above (in the proof of the lemma).

Lemma 3. In each of the closed algebraic sets R_{b,e} there is an irreducible component Y_{b,e} such that the sequence of irreducible cycles {Y_{b,e}|2 ≤ b ≤ d_{k} − 1, 0 ≤ e ≤ w(b)} satisfies the following properties:

(i) Y_{2,0} = R,

(ii) Y_{b,e} ∋ x and, in particular,

codim_{V} Y_{b,e} = k + 1 + ∑_{j=2}^{b−1} w(j) + e,

(iii) for e ≤ w(b) − 1

\frac{mult_{x}}{deg} Y_{b,e+1} ≥ \frac{b + 1}{b} · \frac{mult_{x}}{deg} Y_{b,e}.  (7)

Proof. Assume that the cycles Y_{α,β}, 2 ≤ α ≤ b − 1, and Y_{b,β}, 0 ≤ β ≤ e ≤ w(b) − 1, have already been constructed. Obviously,

O_{V} (D_{b,e+1}) ≅ O_{V} (b)

in the sense of the inclusion V ↪ P, and by (4)

mult_{x} Λ_{b} ≥ b + 1.

Since Y_{b,e} ∋ x is an irreducible cycle, Y_{b,e} ⊄ D_{b,e+1}. Therefore,

\frac{mult_{x}}{deg} (Y_{b,e} • D_{b,e}) ≥ \frac{b + 1}{b} · \frac{mult_{x}}{deg} Y_{b,e}.

Consequently, there is an irreducible component Y_{b,e+1} of the cycle (Y_{b,e} • D_{b,e}), satisfying the inequality (7). Q.E.D. for the lemma.

Corollary. The following estimate holds

\frac{mult_{x}}{deg} R ≤ \frac{3d_{k}}{2d_{k} − 2} · 2^{w^{+}(2)} · ( ∏_{d_{i}≥3} d_{i})^{−1} .

Proof. The last cycle in the sequence Y_{b,e} is an irreducible curve. Its multiplicity at the point x does not exceed its degree. Applying part (iii) of Lemma 3, we get the estimate

\frac{mult_{x}}{deg} R · ∏_{b=2}^{d_{k}−1} (\frac{b + 1}{b})^{w(b)} ≤ 1.

Recalling the definition of the integers w(b) and w^{+}(b), we get:

∏_{b=2}^{d_{k}−1} (\frac{b + 1}{b})^{w(b)} = \frac{2}{3} · \frac{d_{k} − 1}{d_{k}} · ∏_{b=2}^{d_{k}−1} (\frac{b + 1}{b})^{w^{+}(b)} =

= \frac{2d_{k} − 2}{3d_{k}} · 2^{−w^{+}(2)} · (∏_{b=3}^{d_{k}} b^{w^{+}(b−1)−w^{+}(b)}) .

However, w^{+}(b − 1) − w^{+}(b) = ♯{i|1 ≤ i ≤ k, d_{i} = b}. Consequently, the product in brackets is equal to

∏_{d_{i}≥3} d_{i},

which is what we need.

Now everything is ready for the proof of inequality (1). By Lemma 1 codim Bs Λ_{1} = k. This implies that for generic divisors D_{1,e}, e = 1, …, k − 2, the scheme-theoretic intersection

C_{e} = C ∩ D_{1,1} ∩ … ∩ D_{1,e}

is of pure codimension e + 2. Furthermore, all the divisors D_{1,e} are hyperplane sections. Besides, mult_{x} Λ_{1} = 2. Therefore each of the closed sets C_{e} has an irreducible component Y_{e} ∋ x such that

\frac{mult_{x}}{deg} Y_{e+1} ≥ 2\frac{mult_{x}}{deg} Y_{e}

for all e = 1, …, k − 3. Let Y = Y_{k−2} be the last cycle in this sequence. We have

\frac{mult_{x}}{deg} Y ≥ 2^{k−2} · \frac{mult_{x}}{deg} C.

The cycle Y is of the same codimension with respect to V as Bs Λ_{1}. We mentioned above that Bs Λ_{1} is an irreducible cycle by the Lefschetz theorem. Its degree is equal to deg V, and by the regularity condition mult_{x} Bs Λ_{1} = 2^{k}.

If Y = Bs Λ_{1}, then

\frac{mult_{x}}{deg} C ≤ 2^{2−k} · \frac{2^{k}}{deg V} = \frac{4}{deg V} ,

as required.

If Y ≠ Bs Λ_{1}, then there is a divisor D_{1,k−1} ∈ Λ_{1}, which does not contain Y, so that codim_{V} (Y ∩ D_{1,k−1}) = k + 1. Again we take a component R of the cycle (Y • D_{1,k−1}), which satisfies the inequality

\frac{mult_{x}}{deg} R ≥ 2 · \frac{mult_{x}}{deg} Y ≥ 2^{k−1} · \frac{mult_{x}}{deg} C.

The irreducible cycle R is of codimension k + 1. Applying the corollary above to this cycle, we get

\frac{mult_{x}}{deg} C ≤ \frac{3d_{k}}{2d_{k} − 2} · 2^{w^{+}(2)+1−k} · ( ∏_{d_{i}≥3} d_{i})^{−1} .

By definition w^{+}(2) = ♯{i|1 ≤ i ≤ k, d_{i} ≥ 3}. Thus k − w^{+}(2) = ♯{i|1 ≤ i ≤ k, d_{i} = 2}. Finally, we get

\frac{mult_{x}}{deg} C ≤ \frac{3d_{k}}{4d_{k} − 4} · \frac{4}{deg V} .

However, as we mentioned above, d_{k} ≥ 4, so that the first factor in the right-hand part does not exceed one.

Q.E.D. for the inequality (1) and thus for Proposition 1.

2 Regular complete intersections

2.1 Non-regular sets of polynomials

Let us consider the general problem of estimating the codimension of non-regular sets of polynomials, that is, those sets, the zeros of which have a “wrong” dimension.

Let z_{1}, …, z_{N+1} be a set of variables. The symbol P_{a} stands for the space of homogeneous polynomials in the variables z_{∗}. Set

L = ∏_{i=1}^{k+1} P_{m_{i}} = {(p_{1}, …, p_{k+1})}

to be the set all collections of homogeneous polynomials in the variables z_{∗}. Here 0 ≤ k ≤ N − 1 is an arbitrary integer (which is not connected with the number of equations, defining the complete intersection V: in Sec. 2.1 we do not deal with V). With each collection (p_{∗}) ∈ L we associate the projectivized set of its zeros

Z(p_{∗}) = {p_{1} = … = p_{k+1} = 0} ⊂ P^{N} = P(C^{N+1}) = X.

Let

Y = {(p_{∗}) ∈ L| codim_{X} Z(p_{∗}) ≤ k}

be the set of “irregular” collections. Assume that

m = min{m_{1}, …, m_{k+1}} ≥ 2.

Proposition 3. For k ≤ N − 2 the following estimate holds:

codim_{L} Y ≥ mN + m − 1.  (8)

For k = N − 1

codim_{L} Y ≥ min{mN + m − 1, ∑_{i=1}^{N}(m_{i} − 1) + 2}.

Proof. Set

L_{a} = ∏_{i=1}^{a} P_{m_{i}} = {(p_{1}, …, p_{a})}.

For each irregular collection (p_{1}, …, p_{k+1}) fix the first (counting from the left to the right) moment when the codimension of the zero set p_{1} = … = p_{a} = 0 fails to take the correct value. Consider the sets

Y_{a} = {(p_{∗}) ∈ L_{a}| codim_{X} Z(p_{1}, …, p_{a}) = codim_{X} Z(p_{1}, …, p_{a−1}) = a − 1}.

Obviously,

Y = ⋁_{a=1}^{k+1} Y_{a} × L^{a+1},

where we use the notation

L^{a} = ∏_{i=a}^{k+1} P_{m_{i}}.

In particular,

codim_{L} Y = min{codim_{L_{a}} Y_{a}|1 ≤ a ≤ k + 1}.

The inequality (8) does not depend on k. Thus to prove Proposition 3 it is sufficient to show that

codim_{L} Y_{k+1} ≥ mN + m − 1  (9)

for k ≤ N − 2, and

codim_{L} Y_{N} ≥ min{mN + m − 1, ∑_{i=1}^{N}(m_{i} − 1) + 2}.

For any k ∈ {1, …, N − 1} set Y^{∗} = Y_{k+1}. For each collection (p_{1}, …, p_{k+1}) ∈ Y^{∗} the set Z(p_{1}, …, p_{k}) has the correct codimension k, but there exists an irreducible component B ⊂ Z(p_{1}, …, p_{k}), on which p_{k+1} vanishes. Let ⟨B⟩ be the linear span of B, codim⟨B⟩ = b ≤ k. We denote by Y^{∗}(b) the set of all those collections (p_{∗}) ∈ Y^{∗}, for which there exists a component B ⊂ Z(p_{1}, …, p_{k}) such that

codim_{X}⟨B⟩ = b, p_{k+1}|_{B} ≡ 0.

For distinct b ≠ a the sets Y^{∗}(a) and Y^{∗}(b) can have a non-empty intersection. Obviously,

Y^{∗} = ⋃_{b=0}^{k} Y^{∗}(b).

Thus the inequality (9) follows from the estimate

codim_{L} Y^{∗}(b) ≥ mN + m − 1,  (10)

which is what we will actually prove.

The case b = 0. Here ⟨B⟩ = P^{N} and therefore each non-zero monomial in the linear forms in z_{1}, …, z_{N+1} of degree d_{k+1} ≥ d does not vanish on B. The space of monomials

{ ∏_{i=1}^{m_{k+1}}(a_{i,1}z_{1} + … + a_{i,N+1}z_{N+1})} ⊂ P_{m_{k+1}}

is closed. Its dimension is equal to ≥ m(N + 1) − 1 = mN + m − 1. On the other hand, the set of polynomials p_{k+1} ∈ P_{m_{k+1}} that vanish on B is closed. Therefore its codimension is no smaller than ≥ mN + m − 1. This gives (10) for b = 0.

The case b ≥ 1. Here ⟨B⟩ = P^{N−b}. Let us restrict the polynomials p_{i} onto ⟨B⟩. Since B is an irreducible component of the closed set Z(p_{1}, …, p_{k}), one can find (k − b) polynomials among p_{1}, …, p_{k} — let these be p_{1}, …, p_{k−b} — such that there exists a chain of irreducible cycles R_{j} of codimension j in ⟨B⟩, j = 0, 1, …, k − b, R_{0} = ⟨B⟩, B_{j+1} is an irreducible component of the set {p_{j+1} = 0}∩B_{j} and B_{k−b} = B. Consequently, the polynomials

p_{k−b+1}|_{⟨B⟩}, …, p_{k+1}|_{⟨B⟩}

vanish on B. Fix the linear space ⟨B⟩ ⊂ P^{N}. The arguments given above for the case b = 0 show that the condition

(p_{i}|_{⟨B⟩}) |_{B} ≡ 0

for all i ∈ {k − b + 1, …, k + 1} gives not less than

(b + 1)(m(N − b) + m − 1)

independent conditions on (p_{k−b+1}, …, p_{k+1}). Taking into account that the Grassmanian has the dimension dim G(N + 1 − b, N + 1) = b(N + 1 − b), we get finally

codim Y^{∗}(b) ≥ (b + 1)(m(N − b) + m − 1) − b(N + 1 − b)
∥
g(b) + mN + m − 1,

where g(t) = −(m−1)t^{2}+(N(m−1)−2)t is a quadratic function of the real variable t. The function g has exactly two roots, t = 0 and t = γ = N − 2/(m − 1). It is nonnegative on the segment [0, γ]. The parameter b takes integral values 0, …, k ≤ N − 1. This proves (10) in all the cases but one: when m = 2, b = k = N − 1, since only in that case g(b) < 0.

Now we have only one case to consider, that is, when B = ⟨B⟩ is a line in P^{N}, on which vanish all the polynomials p_{1}, …, p_{N}. For a fixed line B the condition p|_{B} ≡ 0 for a polynomial p of degree e ≥ 1 defines a closed algebraic set of polynomials of codimension e + 1 in P_{e}. Therefore,

codim_{L} Y^{∗}(N − 1) ≥ ∑_{i=1}^{N}(m_{i} + 1) − 2(N − 1) = ∑_{i=1}^{N}(m_{i} − 1) + 2,

which is what we need. Q.E.D. for Proposition 3.

2.2 Proof of Proposition 2.

Recall that H stands for the set of all collections (f_{1}, …, f_{k}) of non-zero homogeneous polynomials on P of degrees d_{1}, …, d_{k}, respectively.

To begin with, there is a Zariski open set U_{sm} ⊂ H such that for (f_{1}, …, f_{k}) ∈ U_{sm} the complete intersection V (f_{∗}) is a smooth subvariety of codimension k in P. Furthermore, let us consider the subset

Y = {(x, (f_{∗}))|x ∈ V (f_{∗}) is a non-regular point}

in P × U_{sm}. We claim that the closure

\overline{π(Y)} ⊂ U_{sm}

is a proper closed subset of positive codimension in U_{sm}. Here π: P × U_{sm} → U_{sm} is a projection onto the second factor. Indeed, set

Y (x) = Y ∩ ({x} × U_{sm}) ⊂ P × U_{sm},
I = {(x, (f_{∗}))|x ∈ V (f_{∗})} ⊂ P × U_{sm},
I(x) = I ∩ ({x} × U_{sm}) ⊂ P × U_{sm}.

Identifying {x} × U_{sm} ≅ U_{sm}, we can naturally look at Y (x) and I(x) as subsets in U_{sm}.

Lemma 4. The following estimate holds

codim_{I(x)} Y (x) ≥ M + k + 1.  (11)

Proof. Indeed, the system of equations

q_{1,1} = … = q_{1,k} = 0

defines the tangent space T_{x}V (f_{∗}), which is of dimension exactly M. Projectivizing, we get the set of polynomials

Q = {q_{i,j}|1 ≤ i ≤ k, 2 ≤ j ≤ d_{i}, (i, j) ≠ (k, d_{k})}

of degree ≥ 2 on P^{N}, N = M − 1. Obviously, ♯Q = N. Applying Proposition 3 to the set Q with q_{1,1}, …, q_{1,k} fixed, we see that

codim_{I(x)} Y (x) ≥ min{2M − 1, ∑_{i=1}^{k−1} ∑_{j=2}^{d_{i}}(j − 1) + ∑_{j=2}^{d_{k}−1}(j − 1) + 2}
∥

\frac{1}{2} ∑_{i=1}^{k−1} d_{i}(d_{i} − 1) + \frac{1}{2}(d_{k} − 1)(d_{k} − 2) + 2.

Denote the last expression by ξ = ξ(d_{1}, …, d_{k}). Since M ≥ 2k + 1, the inequality (11) follows from the estimate

ξ(d_{1}, …, d_{k}) ≥ M + k + 1.

Let us compute the minimum of the function ξ in the real arguments (a_{1}, …, a_{k}) ∈ R_{+}^{k}, satisfying the relation a_{1} + … + a_{k} = M + k. Obviously, the minimum is attained at a_{1} = … = a_{k−1} = s ≥ 0, a_{k} = M + k − (k − 1)s ≥ 0. It is easy to compute that

ξ(s, …, s, M + k − (k − 1)s) =

= \frac{1}{2}(k − 1)(ks^{2} − 2(M + k − 1)s) + \frac{1}{2}(M + k − 1)(M + k − 2) + 2 = ζ(s).

The function ζ(s) attains its minimum at s_{∗} = (M + k − 1)/k. This value satisfies all the restrictions. We get

ζ(s_{∗}) = \frac{(M + k − 1)(M − 1)}{2k} + 2.

Since M ≥ 2k + 1, ζ(s_{∗}) ≥ M + k + 1, which is what we need.

Since obviously codim_{U_{sm}} I(x) = k, we get the estimate

codim_{U_{sm}} Y (x) ≥ M + k + (k + 1).

But the set π(Y) is covered by the subsets Y (x) for all x ∈ P, from which we get that

codim_{U_{sm}} \overline{π(Y)} ≥ k + 1,

so that \overline{π(Y)} is a proper subset of U_{sm}. In particular, the open set

U_{reg} = U_{sm} \ \overline{π(Y)}

is non-empty. Q.E.D. for Proposition 2.

2.3 Tangent cones

Let x ∈ X be a non-singular germ, C ∋ x a germ of an irreducible cycle of codimension a ≥ 1,

\begin{aligned}φ: & X̃ & → & X \\ & ⋃ \\ & E & → & x \end{aligned}

the blow-up of the point x, E ≅ P^{dim X−1} the exceptional divisor, C̃ ⊂ X̃ the proper inverse image of C on X̃. The algebraic cycle C̃_{E} = (C̃ • E), the support of which is contained in E, is called the (algebraic) projectivized tangent cone to C at the point x. Denote by π: T_{x}X \ {0} → E the natural projection. The algebraic cycle

\overline{π^{−1}(C̃_{E})},

the support of which lies in T_{x}X ( stands for the closure), is called the (algebraic) tangent cone to C at the point x. We denote it by T_{x}C. Accordingly, the cycle C̃_{E} on E will be also written as P(T_{x}C).

Let D ∋ x be a prime Weil divisor on X. Assume, moreover, that none of the components of the cycle T_{x}D contains a component of the cycle T_{x}C. Then obviously C ⊄ D, (C • D) ⊂ X is a cycle of codimension a + 1. Moreover, we have

Proposition 4. P(T_{x}(C • D)) = (C̃_{E} • D̃_{E})_{E} and thus mult_{x}(C • D) = mult_{x} C · mult_{x} D.

Proof. This is obvious by the general intersection theory [F]. If T_{x}D contains certain components of the cycle T_{x}C, then the multiplicity mult_{x}(C • D) is strictly higher than the product of multiplicities of C and D, see [P2].

If Σ is a non-empty linear system on X and x ∈ Bs Σ, then we define the corresponding tangent linear system T_{x}Σ on T_{x}X ≅ C^{dim X}, setting a generic divisor R ∈ T_{x}Σ to be R = T_{x}D, where D ∈ Σ is a generic divisor. Projectivizing, we get the linear system PT_{x}Σ = Σ̃|_{E}, which consists of projectivized tangent cones PT_{x}D = D̃|_{E}, where D ∈ Σ is a generic divisor. Obviously, T_{x}Σ = \overline{π^{−1}(PT_{x}Σ)}. Note that the tangent linear system consists of cones of degree mult_{x} Σ with the vertex at the point 0 ∈ C^{dim X}.

Defining the tangent linear system, we must take generic, and not arbitrary, divisors D ∈ Σ, because for special divisors, corresponding to subsystems Σ′ ⊂ Σ, their multiplicity at the point x can jump. More exactly, let Σ′ = {D ∈ Σ| mult_{x} D ≥ mult_{x} Σ+1} ⊂ Σ be a proper subsystem of the system Σ. The tangent map T_{x} from Σ to T_{x}Σ is well defined outside Σ′:

\begin{matrix}T_{x}: Σ \ Σ′ → T_{x}Σ, \\ T_{x}: D ↦ T_{x}D. \end{matrix}

Obviously, T_{x} is a usual linear projection from P^{dim Σ} to P^{dim T_{x}Σ}.

In the present paper we consider tangent linear systems for hypertangent linear systems in the sense of Sec. 1.2. Thus we somewhat abuse our terminology, but hope that this will cause no misunderstanding.

Let x ∈ Bs Σ be a base point of the linear system Σ. The following claim is obvious.

Lemma 5. Set-theoretically

T_{x} Bs Σ ⊂ Bs T_{x}Σ.

In particular, in a neighborhood of the point x

codim_{X} Bs Σ ≥ codim_{T_{x}X} Bs T_{x}Σ.

It is this simple fact that we use in the proof of Proposition 1; we need the tangent cones only for this purpose.

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