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Surfaces with Maximal Picard Number Some New Examples Contact Information: Department of Mathematics Partha Solapurkar Purdue University Email: [email protected] Purdue University Webpage: http://www.math.purdue.edu/~psolapur/ explicit calculation of various invariants. We isolate a nice property of elliptic modular Abstract here that there is also a similar example whose fibers are genus 2 curves. surfaces in the following section, which enables us to come up with new examples. We construct some new examples of algebraic surfaces of general type with maximal Picard number. These surfaces arise as certain families of genus 2 or genus 3 curves parametrized by the modular curves and quaternionic Shimura curves. 1 Introduction 4 Hodge Theory of Fibered Surfaces part of a certain weight 2 Hodge structure, as follows: Let f : X → C be a projective such that µ2 = −D, where D > 0 is the discriminant of B. For τ ∈ H, let Λτ denote the rank four lattice η(O) · t(τ, 1) sitting inside C2. Consider the skew-symmetric form the group of divisors modulo algebraic equivalence. Its rank is called the Picard number also assume that f has a section σ : C → X. Let j : U → C be the nonempty Zariski of X, denoted ρ(X). By the Lefschetz theorem on (1, 1)-classes, the first chern class open set over which f : f −1(U ) → U is smooth. We analyze H 2(X, Q) using the Leray spectral sequence. It turns out that all the cohomology classes in the Leray subquotients We say that X has maximal Picard number, or that X is Picard maximal if of H 2(X, C)(1,1) except for those in H 1(C, R1f∗C)(1,1) = H 1(C, j∗j ∗R1f∗C)(1,1) can be represented by linear combinations of fiber components and a section. Thus, we have Theorem. Suppose H 1(C, j∗j ∗R1f∗C)(1,1) = 0. Then X is Picard maximal. We say that f : X → C is extremal if this is the case. 2 An Example of Picard numbers E :O⊗O →Z given by 1 1 E(α, β) = − tr(µαβ̄) = tr(µᾱβ) D D Let Eτ denote the correspondingly defined skew symmetric form on Λτ and also its R-linear extension to C2. Let Aτ denote the complex torus C2/Λτ . By Rotger’s work, certain pure quaternions in O correspond to the elements of NS(Aτ ) and in particular, µ corresponds to the principal polarization Eτ on Aτ . Let Θ denote the universal theta divisor on the family A, defined as the vanishing locus of the Riemann theta function. ρ(X) = h11(X). Our objective is to construct and study examples of such surfaces. ∼ η : B ⊗Q R −→ M2(R). Let O be a maximal order in B and let µ be a pure quaternion of f are connected and that the generic fiber is a nonsingular curve of genus g ≥ 1. We ρ(X) ≤ h11(X). Fix an isomorphism In this section, we reformulate Shioda’s observation in terms of the vanishing of the (1, 1)- Let X be a complex projective surface. The Néron-Severi group of X, denoted NS(X) is always have New Example #2 Let B be an indefinite quaternion division algebra over Q. morphism from a smooth surface to a smooth projective curve. We assume that the fibers map c1 gives an isomorphism between NS(X) ⊗ Q and H 2(X, Q) ∩ H 11(X). Hence we 6 Here’s our reformulation of Shioda’s result: Xn → Cn is extremal, hence Xn is Picard maximal. The theta divisor Θτ on Aτ is either (a) a genus two curve whose Jacobian is Aτ , or (b) consists of two elliptic curves meeting transversely at a point. One can also think of H as the moduli space of principally polarized abelian surfaces Here is an elementary example of Picard numbers of some surfaces that might help one with quaternionic multiplication given by the data (O, µ), together with a symplectic develop a feel for the behavior of the Picard number. Let X = E1 × E2, where E1 and basis for the lattice, and A the universal family over it. E2 are elliptic curves. For such a surface, h11(X) = 4. We have: 5 (a) ρ(E1 × E2) = 2, if E1 and E2 are two nonisogenous elliptic curves. The divisor classes are represented by E1 × 0 and 0 × E2. Our idea is to take the elliptic modular surfaces X2n → C2n and replace every fiber with a canonically constructed higher genus curve. The modular curve U2 is isomorphic (b) ρ(E × E) = 3, if E is an elliptic curve without complex multiplication. The addi- to P1 − {0, 1, ∞}. The family X2 over U2 is the Legendre family of elliptic curves, given tional divisor class is the class represented by the diagonal in E × E. (c) ρ(E × E) = 4, if E is an elliptic curve with complex multiplication. The new divisor class is represented by the graph of an endomorphism E → E that comes from the complex multiplication. In this case, E × E is Picard maximal. This suggests that surfaces with maximal Picard number are very special surfaces. 3 Elliptic Modular Surfaces Let H denote the upper half plane, and let E → H be the universal family of elliptic curves E together with a symplectic basis for H 1(E, Z). If instead, we keep track of a symplectic basis of H 1(E, Z/nZ) for n ≥ 2, we obtain a universal family En of elliptic curves over the noncompact algebraic modular curve Un. We can compactify this family New Example #1 Let O1 denote the group of elements of O with reduced norm 1. The subgroup η(O1) of SL2(R) acts on H and by Shimura’s work the quotient H/η(O1) is a compact Riemann surface. We consider certain quotients of Θ → H by an appropriate choice of a finite index subgroup O1. For example, one can take G to be the full level 8n subgroup by y 2 = x(x − 1)(x − t) for t ∈ P1 − {0, 1, ∞}. Let f : X → P 1 be obtained by blowing up P2 along the base locus of the pencil {x4 + y 4 + z 4 + t(x2y 2 + y 2z 2 + z 2x2) = 0 : t ∈ P1} These curves are nonsingular when t 6∈ S = {−1, ±2, ∞}. The curve Et = Xt/{x 7→ ±x} can be identified with the elliptic curve y 2 = x(x − 1)(x + t + 1), and the map induces an isogeny Pic0(Xt) ∼ Et3. This is actually an isogeny of abelian schemes from Pic0(X/P1 − S) to L|P1−S , where g : L → P1 − S is the pullback of the Legendre family along the automorphism of P 1 that fixes 0, 1 and sends t 7→ −(t + 1). Hence R1f∗Q = R1g∗Q⊕3, in a suitable way to get a compact algebraic surface Xn over the compact modular curve and thus f : X → P1 is extremal. Now, C2n comes with a natural map onto C2 = P1, which is unramified over U2. One can show that the pull back X ×C2 C2n → C2n is also Cn. Shioda observed that elliptic modular surfaces have maximal picard number, by extremal. Hence X ×C2 C2n is a Picard maximal surface for every n ≥ 1. We mention of Sp(4, Z). Let f : X → C be such a quotient. Then the generic fiber of f is a genus two curve, and the singular fibers of f are unions of two elliptic curves meeting transversely at one point. Theorem. f : X → C is extremal, and hence X is Picard maximal. The proof essentially boils down to verifying that the Kodaira-Spencer class κ for the family Θ → H induces an isomorphism ∼ H 0(Θτ , ΩΘτ ) −→ H 1(Θτ , OΘτ ). Acknowledgements I would like to thank my advisor Professor Donu Arapura for guidance and help at every stage of this research project.