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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.