On Compact Shimura Surfaces

Håkan Granath Max-Planck-Institut für Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany email: granath@mpim-bonn.mpg.de

Abstract

We study surfaces constructed from groups of units in quaternion orders Λ over the integers in real quadratic fields k. A short presentation of some general theory of such surfaces is given, in particular, we construct certain “modular curves” on the surfaces and examine their properties. We apply our results to some surfaces related to the case k = ℚ(\sqrt{13}) and d(Λ) = (3), and find their place in the Kodaira classification of algebraic surfaces.

Key words: Shimura surface, quaternion order, Kodaira classification 2000 Mathematics Subject Classification: 14G35, 11G18.

The purpose of this paper is to describe some tools to study particular examples of quaternionic Shimura surfaces from the point of view of algebraic geometry. To do that one needs of course standard numerical invariants, but also more explicit knowledge of the geometry of the curves on these surfaces. In section 2, we construct a family of “modular curves”, and state some results regarding these. Proofs of all claims can be found in [5]. In section 3 we consider an explicit example over the base field ℚ(\sqrt{13}). For more details about the underlying computations, we refer to [5].

1 Introduction

Let k = ℚ(\sqrt{d}) be a real quadratic field with R = O_{k} and D = d(k). Let A be a totally indefinite quaternion algebra over k which allows an involution of type 2, i.e. a map A → A, denoted x ↦ \overline{x}, such that \overline{\overline{x}} = x, \overline{x + y} = \overline{x} + \overline{y} and \overline{xy} = \overline{x} \overline{y} for all x, y ∈ A, and such that the restriction of the involution to k is the non-trivial automorphism of k. If Λ is a maximal R-order in A, then we let Λ_{0} = \{λ ∈ Λ | \overline{λ} = λ\}. It can be shown that, replacing the involution of type 2 with another one if necessary, one can assume that D and d(Λ_{0}) are coprime. In that case, we say that the involution is special with respect to Λ.

The canonical involution of a quaternion algebra A is denoted x ↦ x^{∗}. The reduced norm is nr(x) = xx^{∗} and the reduced trace is tr(x) = x + x^{∗}. The reduced discriminant of an order Λ is denoted d(Λ), and we let d_{0}(Λ) be the positive rational integer that generates d(Λ) if such exists.

Choose a maximal order Λ and let Λ^{1} = {λ ∈ Λ | nr(λ) = 1}. Choose a real representation A → M_{2}(ℝ), and consider the induced action of Λ^{1} on H given by Möbius transformations. We get an action on H×H by λ(z_{1}, z_{2}) = (λz_{1}, \overline{λ}z_{2}). We let Γ denote the image of Λ^{1} in Aut(H×H) and X = H×H/Γ.

X is a so called Hilbert modular surface if A ≅ M_{2}(k). These are not compact, but can be compactified by adding a finite number of points, the so called cusps. These points give complicated singularities which were first resolved by Hirzebruch, see [9]. Studying numerical invariants of the surfaces and configurations of certain curves on them, curves coming from the resolution of the singularities and so called modular curves, one managed to identify many of these surfaces in terms of the Kodaira classification of algebraic surface. For example, if Λ = M_{2}(R), then X is rational if and only of D = 5, 8, 12, 13, 17, 21, 24, 28, 33, or 60 (see [8]). See also [4].

From now on we will consider non-split algebras, i.e. we assume that A ≅ M̸_{2}(k). In this case the quotient X = H×H/Λ^{1} is compact. However, in general X has quotient singularities coming from elements of finite order in Γ, so called elliptic elements. We let Y denote the canonical minimal resolution of X. We now summarise some known results about the numerical invariants of X and Y in this case. Let ω be the Gauss-Bonnet form on H×H:

ω = \frac{1}{(2π)^{2}} \frac{dx_{1} ∧ dy_{1}}{y_{1}^{2}} ∧ \frac{dx_{2} ∧ dy_{2}}{y_{2}^{2}} ,

where z_{k} = x_{k} +iy_{k}, k = 1, 2 are the standard coordinates on the two factors of H×H. The Euler characteristic e(X) is given by

e(X) = ∫_{F} ω + ∑_{r>1} e_{r} \frac{r − 1}{r},  (1)

where F is a fundamental domain of the group action and e_{r} is the number of classes of elliptic points of order r (see [6], p. 197). The formula for the volume of a fundamental domain in H×H under the action of Γ can be found for example in [12], p. 193. In our case, when Λ is a maximal order and the quaternion algebra A admits an involution of type 2, the formula may be written as

∫_{F} ω = 2ζ_{k}(−1) ∏_{p|d_{0}(Λ)}(p − 1)^{2}.  (2)

Here ζ_{k} is the zeta-function of the field k. The value ζ_{k}(−1) can be easily computed using a formula of Siegel (see e.g. [6], p. 192). By Satz 8 in [3], we have that the irregularity q of the surface Y vanishes: q(Y) = 0. We also have, since A is a skew field, that χ(X) = \frac{1}{4}e(X) (see [12], p. 206–207).

Apart from the general facts above, not very much is known about the geometry of specific examples. In [10] all cases were determined when X is non-singular and has geometric genus p_{g} = 0. There are 3 such cases, one with k = ℚ(\sqrt{2}) and two with k = ℚ(\sqrt{3}). One can also find there results about the corresponding question for certain extensions of the group Γ. This gives many more cases with p_{g} = 0. Note however, that in none of these cases the algebra allows an involution of type 2.

Notation. For p a prime, we let v_{p} denote the valuation on ℚ_{p} and (a, b)_{p} the Hilbert symbol, a, b ∈ ℚ_{p}^{∗}. For integers a, b, c, we let φ = [a, b, c] denote the quadratic form ax^{2} +bxy+cy^{2}. The discriminant of φ is d(φ) = b^{2} −4ac, and the content m(φ) of φ is the ideal generated by a, b and c.

2 The modular curves

Consider the following 4-dimensional ℚ-vector space:

W = \{β ∈ A | \overline{β}^{∗} = β\}.

Clearly, if β ∈ W, then nr(β) ∈ ℚ. If β ∈ W and nr(β) > 0, then we define the following curve on H×H:

C_{β} = {(z, βz) | z ∈ H}.

This curve maps to a subvariety F_{β} ⊂ X.

We let Γ_{β} = \{λ ∈ Λ^{1} | λ · C_{β} = C_{β}\} = \{λ ∈ λ^{1} | βλ = ±\overline{λ}β\}, the stabiliser group of C_{β}. We define A_{β} = \{a ∈ A | βa = \overline{a}β\}, which is a quaternion algebra over ℚ, and let Λ_{β} = A_{β} ∩ Λ. We have that Λ_{β}^{1} is a subgroup of Γ_{β} of index 1 or 2.

If we consider A as a right A-module, then we say that a map Φ : A × A → A is an A-hermitian form if Φ(x + y, z) = Φ(x, z) + Φ(y, z), Φ(xa, y) = Φ(x, y)a and Φ(x, y) = \overline{Φ(y, x)}^{∗} for all a, x, y, z ∈ A. If Λ is a maximal order in A and the type 2 involution on A is special with respect to Λ, then we say that the A-hermitian form Φ is integral with respect to the order Λ if

i) Φ(x, y) ∈ Λ for all x, y ∈ Λ,

ii) Φ(x, x) ∈ R + \sqrt{D}Λ for all x ∈ Λ.

For any β in W, we define a hermitian form Φ_{β} by Φ_{β}(x, y) = \overline{y}^{∗}βx. We define a ℤ-lattice L of rank 4 by

L = {β ∈ W | Φ_{β} is integral}.  (3)

Consider a quadratic form q on L by q(β) = \frac{d_{0}(Λ)}{d_{0}(Λ)} nr(β). We define a lattice

L^{#} dual to L by

L^{#} = {l ∈ W | tr(l^{∗}L) ⊆ ℤ},  (4)

and a quadratic form q^{#} on L^{#} given by q^{#}(l) = Dd_{0}(Λ_{0}) nr(l). One can show that (L, q) and (L^{#}, q^{#}) are primitive integral quadratic forms, and that their genera are independent of the choice of involution.

Consider now the quaternary quadratic lattice (L^{#}, q^{#}). Recall the definition of the even Clifford algebra C_{0}(L^{#}, q^{#}) as the quotient of the tensor algebra T_{0}(L^{#}) = ⨁_{k=0}^{∞}(L^{#})^{⊗2k} by the ideal generated by all elements x ⊗ x − q^{#}(x), where x ∈ L^{#}. Define a function ϕ : L^{#} ⊗_{ℤ} L^{#} → A by

ϕ(l_{1} ⊗ l_{2}) = Dd_{0}(Λ_{0})l_{1}^{∗}l_{2}.

We can extend ϕ in a natural way to the even tensor algebra, so we get a map ϕ : T_{0}(L^{#}) → A. This map clearly vanishes on elements of the form x ⊗ x − q^{#}(x), where x ∈ L^{#}, giving an embedding of the ring C_{0}(L^{#}, q^{#}) into the algebra A. In fact, the image is a subring (but not a suborder) of Λ:

ϕ : C_{0}(L^{#}, q^{#}) → Λ.  (5)

The map (5) is one of the main ingredients in the proofs of theorem 2 and theorem 7 below.

Combining the following two results, the genus of the orders Λ_{β} is determined.

Proposition 1. (Cf. prop. 1.12 in [2]) Let F be a local field, K a separable maximal commutative subalgebra of A and S a maximal order of K. Let Λ_{1}, Λ_{2} ⊂ A be two S-primitive orders (i.e. they allow an embedding of S). If d(Λ_{1}) = d(Λ_{2}), then Λ_{1} ≅ Λ_{2}.

Theorem 2. (See [5], thm. 7.16 and prop. 7.23) The discriminant of Λ_{β} equals the least common multiple of q(β) and d_{0}(Λ). Furthermore, if p is a prime, then the following holds:

i ) If p is split in k, then (A_{β})_{p} is split if and only if p ∤ d(Λ). If p is unramified in k, then (A_{β})_{p} is split if and only if v_{p}(q(β)) is even. If p is ramified in k, then (A_{β})_{p} is split if and only if nr(β) ∈ nr_{k_{p}/ℚ_{p}}(k_{p}^{∗}).

ii ) The order (Λ_{β})_{p} is R_{p}-primitive unless p is split in k and p | d(Λ).

An element β ∈ L is called primitive if {x ∈ ℚ | xβ ∈ L} = ℤ. For N a positive integer, let L(N) denote the set of primitive elements of L with q(β) = N.

Proposition 3. (See [5], prop. 8.8) If β ∈ L(N), then [Γ_{β} : Λ_{β}^{1}] = 2 if and only if the following conditions hold for every prime p such that p | D : i ) if (−1, D)_{p} = 1, then p | N,

ii ) if (−1, D)_{p} = −1, then v_{p}(d) ≤ v_{p}(N) < 2v_{p}(D).

By proposition 3, there exists a function κ defined on the set of all positive integers primitively represented by (L, q), such that [Γ_{β} : Λ_{β}^{1}] = κ(N) for all β ∈ L(N). For every positive integer N, we define

F_{N} = ⋃_{β∈L(N)} F_{β}.

Let f_{N} denote the number of irreducible components of F_{N}. For every prime p dividing D, we define a character χ_{D,p} : ℤ_{p} → {±1, 0}, by χ_{D,p}(N) = (D, N)_{p} if p ∤ N and χ_{D,p}(N) = 0 if p | N. Furthermore, if N ∈ ℤ_{p} and N ≠ 0, then we define a_{D,p}(N) = 2 if v_{p}(N) ≥ 2v_{p}(D) and a_{D,p}(N) = 1 otherwise.

Theorem 4. (See [5], thm. 8.9) If there exists a prime p such that p | d(Λ) and p^{2} | N, then f_{N} = 0. Otherwise

f_{N} = \frac{κ(N)}{2} ∏_{p|D}(χ_{D,p}(NB) + 1)a_{D,p}(N),

where B = d_{0}(Λ_{0})/d_{0}(Λ).

Now we have the tools to determine the number of components of F_{N} and their genus. We turn to the problem of determining how these curves intersect. If z = (z_{1}, z_{2}) is a point in H×H, then we let

L_{z} = {β ∈ L | q(β) > 0 and βz_{1} = z_{2}} ∪ {0}.

We have that q restricted to L_{z} is positive definite, so the rank of L_{z} is at most 2. The point z is called special if the lattice L_{z} has rank 2. It is possible to choose an orientation on the 2-dimensional ℝ-vector bundle L = {(z, β) ∈ (H×H) × M_{2}(ℝ) | det(β) > 0 and βz_{1} = z_{2}} ∪ H×H × {0} on H×H. We fix such a choice, and get induced orientations on the lattices L_{z}. Let q_{z} = q|_{L_{z}} be the oriented binary form associated with the special point z.

We introduce (as in [7]) a function s counting the number of special points. Let φ be an oriented binary form. Then we put

s(φ) = ∑_{ζ∈X(φ)} \frac{w_{φ}}{v_{ζ}} ,  (6)

where w_{φ} is the number of elements in the group of oriented automorphisms of φ, and v_{ζ} is the order of the isotropy group Λ_{z}^{1} in Λ^{1} of a point z ∈ H×H representing ζ (so w_{φ} and v_{ζ} both take values in the set {2, 4, 6}).

From now on, we will assume that (D, 6) = 1. In this case, every elliptic point is of order 2, 3 or 5 (the latter case is only possible if k = ℚ(\sqrt{5})), and one can show that every elliptic point of order 2 or 3 is a special point.

Proposition 5. (See [5], cor. 9.9) Let q_{z} be the quadratic form of an elliptic point z of order n = 2 or n = 3. There are two cases:

i ) q_{z} ≅ mDφ, where φ = [1, 0, 1] if n = 2 or φ = [1, −1, 1] if n = 3, and where m is a positive integer with m | d(Λ). If n = 2, then we have Λ_{z}^{1} ∩ Λ_{β}^{1} = {±1} and Λ_{z}^{1} ⊆ Γ_{β} for all β ∈ L_{z} \ {0}, and if n = 3, then we have Λ_{z}^{1} ∩ Γ_{β} = {±1} for all β ∈ L_{z} \ {0}.

ii ) q_{z} ≅ mφ, where d_{0}(φ) = −4D if n = 2 or d_{0}(φ) = −3D if n = 3, and m is a positive integer with m | d(Λ). We have Λ_{z}^{1} ⊆ Λ_{β}^{1} for all

β ∈ L_{z} \ {0}.

Lemma 6. (See [5], lem. 9.5) Let p be a prime and φ_{p} a non-degenerate binary quadratic form over ℤ_{p}. Then (L_{p}, q_{p}) represents φ_{p} primitively if and only if the following conditions are satisfied:

i ) if p is unramified in k, then p ∤ m(φ_{p}),

ii ) if p | d(Λ), then φ_{p} is anisotropic, and it is either a modular form with p^{2} ∤ m(φ_{p}) or a non-modular form such that d(φ_{p}) = (p) if p ≠ 2 and such that d(φ_{p}) | 8 and φ_{p} do not primitively represent 4 if p = 2,

iii ) if p | D, then p | d(φ_{p}) and p^{2} ∤ m(φ_{p}). Furhermore, if m(φ_{p}) = (1), then φ_{p} represents B = d_{0}(Λ_{ℤ})/d_{0}(Λ), and if m(φ_{p}) = (p), then φ_{p} represents −BD.

If φ is a positive definite oriented binary form such that φ ≅ q_{z} for some point z ∈ H×H, then we let Δ = d(φ), and m > 0 be a generator of m(φ). We write m = m_{1}m_{2}m_{3}, where m_{1} contains those prime factors which divide D, m_{2} those prime factors which divide d(Λ) and m_{3} contains all other prime factors. By lemma 6, we have that m_{1} | D, that m_{2} | d(Λ) and that every prime dividing m_{3} is split in k. If φ is a binary form over ℤ and p a prime, then we let α_{p}(φ) = 1 if φ_{p} is modular, and α_{p}(φ) = 2 otherwise. We also introduce the modified class number h′ (compare e.g. [7]), which counts primitive binary forms φ with a given discriminant N < 0 with multiplicity 2/w_{φ}. Hence h′(−3) = 1/3, h′(−4) = 1/2 and h′(N) = h(N) otherwise.

Theorem 7. (See [5], thm. 9.16) A positive definite oriented binary form φ is represented by q if and only if φ_{p} satisfies the conditions of lemma 6 for every prime p. For such a form, we have

s(φ) = 2^{a−1}h′ (\frac{Δ}{m_{2}^{2}D} ) ∏_{p|D} α_{p}(φ) ∏_{p|d(Λ)} \frac{2}{α_{p}(φ)},  (7)

where a is the number of different primes dividing m_{3}.

Theorem 7 is our tool to determine intersection points of the modular curves. It remains to determine the self-intersections of the curves F_{β}. For that one can use the theory of Chern divisors (see [6], p. 252).

3 Some surfaces over ℚ(\sqrt{13})

We will now consider the example where k = ℚ(\sqrt{13}) and d_{0}(Λ) = 3. This is one of the cases with relatively small hyperbolic volume of the fundamental domain. Since h(k) = 1 in this case, there is only one maximal order in A up to isomorphism (by Eichler’s norm theorem and lemma III.5.6 in [11]).

We consider first discrete groups of Aut(H×H) extending Γ. Note that every element in A^{+} = {a ∈ A | nr(a) is totally positive} acts on H×H, giving a map ϱ : A^{+} → Aut(H×H). The totally positive elements of the normaliser of Λ,

N^{+}(Λ) = {n ∈ A^{+} | nΛn^{−1} = Λ},

naturally act on X. Let Γ_{II} = ϱ(N^{+}(Λ)). In our case, we have that Γ_{II} is given as follows: Let v = 4 + \sqrt{13}, so 3 = v\overline{v}. There exists λ_{x} ∈ Λ such that nr(λ_{x}) = x for x = v, \overline{v} and 3, and we let γ_{x} = ϱ(λ_{x}) : H×H → H×H. This gives three well defined involutions on X, which we denote ι_{3}, ι_{v} and ι\overline{v}. We define Γ_{I} = Γ ∪ γ_{3}Γ, and we have in fact that Γ_{II} = Γ_{I} ∪ γ_{v}Γ_{I}.

We will now define an extension of Γ_{II}. Let us fix an element β_{13} ∈ L with q(β_{13}) = 13, and define a map T_{0} : H×H → H×H, by

T_{0}(z_{1}, z_{2}) = (β_{13}^{∗}z_{2}, β_{13}z_{1}).

We have that T_{0}^{2} is the identity, and one can check that T_{0}Γ = ΓT_{0}, so T_{0} induces an involution on X, which we denote T. We have T ι_{x} = ι\overline{x}T, for x ∈ \{1, v, \overline{v}, 3\}. Let Γ_{III} = Γ_{II} ∪ T_{0}Γ_{II}. This group is independent of the choice of the element β_{13}. Hence, we have a tower of discrete groups acting on H×H

Γ ⊂ Γ_{I} ⊂ Γ_{II} ⊂ Γ_{III}.  (8)

We will consider the 4 quotient surfaces X, X_{I} = H×H/Γ_{I}, X_{II} = H×H/Γ_{II} and X_{III} = H×H/Γ_{III}. We let Y, Y_{I}, Y_{II} and Y_{III} respectively denote the canonical minimal resolution of the corresponding quotient surface. We let G_{I} = \{ι_{1}, ι_{3}\}, G_{II} = \{ι_{1}, ι_{v}, ι\overline{v}, ι_{3}\} and G_{III} = G_{II} ∪ T G_{II} denote the corresponding groups acting on X, so G_{III} ≅ D_{4}. We have that G_{I} is the center of G_{III}, so the filtration (8) is in some sense canonical. Our main result is:

Theorem 8. (See [5], thm. 12.1) Y is a minimal surface of general type, Y_{I} is a K3-surface blown up 4 times, Y_{II} is an Enriques surface blown up 2 times and Y_{III} is a rational surface with Euler number e = 12.

Calculation of the quadratic form q in this case gives:

q = 2t_{0}^{2} + t_{0}t_{1} + 5t_{1}^{2} − 13(t_{2}^{2} − t_{2}t_{3} + t_{3}^{2})  (9)

By theorem 4, the number of components of F_{N} is

f_{N} = \begin{cases}0 & if (\frac{N}{13}) = 1 or 9 | N, \\ 2 & if 13^{2} | N and 9 ∤ N, \\ 1 & otherwise. \end{cases}  (10)

If β is a primitive element of L, then we have, by theorem 2, that d_{0}(Λ_{β}) is the least common multiple of q(β) and 3. By proposition 3, we have

κ(N) = \begin{cases}2 & if 13 | N, \\ 1 & otherwise. \end{cases}

The action of G_{III} on the curves are given by

Lemma 9. (See [5], lem. 12.4) For every positive integer N represented by q, we have that ι_{3}(F_{N}) = F_{N} and T (F_{N}) = F_{N}. If 3 ∤ N, then ι_{x}(F_{N}) = F_{3N} and ι_{x}(F_{3N}) = F_{N}, for x = v and x = \overline{v}.

Now, X has 8 elliptic points of order 2, and 4 of order 3, corresponding to elliptic elements ϵ_{1}, …, ϵ_{8} and ρ_{1}, …, ρ_{4} respectively. The elements of order 2 give rise to (−2)-curves on Y, ρ_{1}, ρ_{2} to two intersecting (−2)-curves each and ρ_{3}, ρ_{4} to (−3)-curves. In addition to these curves, we will consider the 6 irreducible curves F_{2}, F_{6}, F_{5}, F_{15}, F_{13} and F_{39}. The binary forms associated to the elliptic points, and the branches of the above curves that pass through them, are given by table 1. All other intersection points of these curves are given in table 2. We get that F_{2} and F_{6} are non-singular rational curves, F_{13} and F_{39} are non-singular elliptic curves and F_{5} and F_{15} are elliptic curves with two nodes each.

Determining the fixed points of the elements of G_{III} requires explicit calculations. The isotropy groups of all points with non-trivial isotropy group Γ_{II,z} are given in table 3. F_{13} and F_{39} are pointwise fixed by T and T ι_{3} respectively. All other fixed points of elements in G_{II}T are already fixed points of elements of G_{II}.

We now present the configurations of curves on the different surfaces. We start with the surface Y and proceed by a sequence of blow ups and quotient constructions. We will use the convention that whenever we have two surfaces Z and Z̃, then Z̃ is isomorphic to the surface Z blown up in a finite set of points. For easier navigation among the surfaces, we first

Elliptic element Order Binary form  Branches
ϵ_{1}, ϵ_{2}  2  [13, 0, 13]  F_{13}, F_{13}
ϵ_{3}, ϵ_{4}  2  [39, 0, 39]  F_{39}, F_{39}
ϵ_{5}, ϵ_{6}  2  [2, 2, 7]  F_{2}
ϵ_{7}, ϵ_{8}  2  [6, 6, 21]  F_{6}
ρ_{1}, ρ_{2}  3  [13, −13, 13]  F_{13}, F_{39}
ρ_{3}  3  [2, 1, 5]  F_{2}, F_{6}, F_{5}, F_{15}
ρ_{4}  3  [2, −1, 5] F_{2}, F_{6}, F_{5}, F_{15}

Table 1: Elliptic points

Binary form Branches  s(φ)
[2, 0, 39]  F_{2}, F_{39}  h(−24) = 2
[6, 0, 13]  F_{6}, F_{13}  h(−24) = 2
[5, −3, 5]  F_{5}, F_{5}, F_{13}  2h(−7) = 2
[15, −9, 15] F_{15}, F_{15}, F_{39} 2h(−7) = 2
[5, 2, 8]  F_{5}, F_{15}  h(−12) = 1
[5, −2, 8]  F_{5}, F_{15}  h(−12) = 1
[5, 0, 39]  F_{5}, F_{39}  h(−60) = 2
[15, 0, 13]  F_{15}, F_{13}  h(−60) = 2
[5, −5, 11]  F_{5}, F_{39}  h(−15) = 2
[7, −1, 7]  F_{15}, F_{13}  h(−15) = 2
[13, 0, 39]  F_{13}, F_{39}  h(−156)/2 = 2

Table 2: Intersection points

Γ_{z}  Γ_{I,z} Γ_{II,z}  Γ_{III,z}
[2, −2, 7]  ℤ/2 } ℤ/2 }
[2, −2, 7] ℤ/2 ℤ/2 ℤ/4
[6, −6, 21]  ℤ/2 } ℤ/2
[6, −6, 21]  ℤ/2
[13, 0, 13]  ℤ/2 } ℤ/2 }
[13, 0, 13] ℤ/2 ℤ/2 ℤ/2 × ℤ/2
[39, 0, 39]  ℤ/2 } ℤ/2
[39, 0, 39]  ℤ/2
[2, 1, 5]  ℤ/3 ℤ/6 } ℤ/6  ℤ/12
[2, −1, 5]  ℤ/3 ℤ/6
[13, −13, 13] ℤ/3 ℤ/6 } ℤ/6  D_{6}
[13, −13, 13] ℤ/3 ℤ/6
[13, 0, 39]  (1) ℤ/2 } ℤ/2 ℤ/2 × ℤ/2
[13, 0, 39]  (1) ℤ/2
[5, 2, 8]  (1) ℤ/2 } ℤ/2  ℤ/4
[5, −2, 8]  (1) ℤ/2

Table 3: Overview of points z with non-trivial isotropy group Γ_{II,z}

summarise in a diagram the sequence of surfaces that we will construct:

\xymatrix{ & Ỹ \ar[dl]_{blow up} \ar[dr]^{2:1} \\ Y &  & Y_{I} \ar[dr]^{2:1} &  & Ỹ_{II} \ar[dl]^{blow up} \ar[dr]^{2:1} \\  &  &  & Y_{II} &  & Ỹ_{III} \ar[dr]^{blow down} \\  &  &  &  &  &  & Y_{III}}  (11)

We have drawn the configuration of curves on Y in figure 1. Here all curves drawn with thick lines are exceptional curves coming from the canonical resolution of the singularities of X. If the self-intersection number of such a curve is not explicitly given in the figure, then it is −2. Furthermore, the rational curves F_{2} and F_{6} have self-intersection −2 and the elliptic curves F_{13} and F_{39} have self-intersection −4. The nodal elliptic curves F_{5} and F_{15} have self-intersection 2.

The involution ι_{3} has 8 fixed points on X. On the surface Y, the fixed point locus of the lifted involution consists of the two (−3)-curves and the 10

F_{2} F_{6} F_{39} F_{13}  F_{5} F_{15} x_{1}  x_{2}  x_{3}  x_{4}  x_{5}  x_{8} x_{6} x_{7}  x_{9} x_{10} −3 B_{1}  −3 B_{2} A_{1}  A_{2} A_{3}  A_{4} Figure 1: Y

isolated points x_{1}, …, x_{10}, which are indicated in figure 1. Blowing up these 10 points on Y to the surface Ỹ, and thereafter taking the quotient, we get a surface which, in fact, is the minimal desingularisation Y_{I} of X_{I}. We indicate the configuration of curves in figure 2. We have that F_{2}^{2} = F_{6}^{2} = −1, that F_{13} and F_{39} are non-singular rational curves with F_{13}^{2} = F_{39}^{2} = −4, and that F_{5} and F_{15} are nodal rational curves with F_{5}^{2} = F_{15}^{2} = 0.

We let ι_{I} denote the involution on Y_{I} induced by the group Γ_{II}. It is fixed point free. If N is not divisible by 3, then ι_{I} maps F_{N} to F_{3N} and vice versa. We denote their common images in Y_{II} by F_{N}. We have of course that the self-intersection of F_{2}, F_{13} and F_{5} respectively, are the same as for the corresponding curves on Y_{I}. The configuration on Y_{II} is drawn in figure 3.

The involution on Y_{II} induced by Γ_{III} has the curve F_{13} and the 6 isolated points y_{1}, …, y_{6} as fixed point locus. We blow up these points and then we take the quotient. The surface we get is Y_{III} blown up 5 times, see figure 4. On this surface, we have that F_{2}, F_{13} and F_{5} are non-singular rational curves and we have F_{2}^{2} = −1, F_{13}^{2} = −8 and F_{5}^{2} = 0.

We blow down the exceptional curve configurations which meet F_{13} and we get the surface Y_{III}. See figure 5.

Since ζ_{k}(−1) = \frac{1}{6}, we get that ∫_{X} ω = \frac{4}{3} by equation (2). Furthermore e_{2} = 8 and e_{3} = 4, so by (1), we get e(X) = 8. It is now straightforward to compute the numerical invariants of all surfaces. We summarise the result in the following table:

F_{2} F_{6}  F_{39} F_{13} F_{15} F_{5} −6  −6 A_{1} A_{2} Figure 2: Y^{I}

F_{13} F_{2}  F_{5} A  D −6  y^{1} y^{2}  y_{3} y_{4} y_{5} y_{6}

Figure 3: Y_{II}

F_{13} F_{2}  F_{5} −1 −2  −2 −2  −2 −2  −2  −1 −4  −2  −2  −1 −2 −2

Figure 4: Ỹ_{III}

F_{13} F_{2}  F_{5} −2  −2 −2  −2 −2  −2 −4  −2  −2

Figure 5: Y_{III}

Y Y_{I} Y_{II} Y_{III}
e 22 28 14 12
χ 2 2 1 1
K^{2} 2 −4 −2 0
q 0 0 0 0
p_{g} 1 1 0 0

We have the following corollary of Castelnuovo’s criterion for rationality of a surface (see [6], corollary 1, p. 255):

Proposition 10. Let Z be a non-singular surface with q = 0. If Z contains a non-singular rational curve D with D^{2} ≥ 0, then Z is rational.

This directly applies to give that Y_{III} is a rational surface. Namely, if we blow down the connected configuration consisting of F_{2} and three (−2)- curves, then the image of the (−4)-curve is a non-singular rational curve with self-intersection 0.

Studying the intersection matrix of the divisors generated by the curves in figure 4, we find that these curves generate a sublattice of Pic(Ỹ_{III}) of index 2. Using the adjunction formula, one gets the following representation of the canonical divisor of Ỹ_{III}:

2K = 4F_{2} − F_{13} + 3C_{14} + 2C_{15} + C_{16}.  (12)

Using (12), we can trace 2K backwards and get the following relation on Y_{II}:

2K = 4F_{2} + 2A.  (13)

Let now Z denote the surface we get if we blow down first F_{2} and then A on Y_{II}. By (13), we have that 2K = 0 on Z. Furthermore, e(Z) = 12 and q(Z) = 0. This shows that Z is an Enriques surface (see [1], chapter VI). Since the map Y_{I} → Y_{II} is unramified, we immediately get that Y_{I} is a K3-surface blown up 4 times with

K = 2F_{2} + A_{1} + 2F_{6} + A_{2}.  (14)

This implies that the canonical divisor on Y is given by

K = 2F_{2} + 2F_{6} + ∑ A_{i} + ∑ B_{i}.

We conclude that Y is a minimal surface (since if E an exceptional divisor on Y, then KE = −1, which is impossible since K is effective and does not contain any exceptional component). Furthermore, since K is effective and K^{2} > 0, we have that Y is of general type (see [1]).

References

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