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Register a new userA Hilbert Irreducibility Theorem for Enriques surfaces
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We introduce a novel approach to Bertini irreducibility theorems over an arbitrary field, based on random hyperplane slicing over a finite field. Extending a result of Benoist, we prove that for a morphism $phi colon X to mathbb{P}^n$ such that $X$ is geometrically irreducible and the nonempty fibers of $phi$ all have the same dimension, the locus of hyperplanes $H$ such that $phi^{-1} H$ is not geometrically irreducible has dimension at most $operatorname{codim} phi(X)+1$. We give an application to monodromy groups above hyperplane sections.
Given two semistable, non potentially isotrivial elliptic surfaces over a curve $C$ defined over a field of characteristic zero or finitely generated over its prime field, we show that any compatible family of effective isometries of the N{e}ron-Severi lattices of the base changed elliptic surfaces for all finite separable maps $Bto C$ arises from an isomorphism of the elliptic surfaces. Without the effectivity hypothesis, we show that the two elliptic surfaces are isomorphic. We also determine the group of universal automorphisms of a semistable elliptic surface. In particular, this includes showing that the Picard-Lefschetz transformations corresponding to an irreducible component of a singular fibre, can be extended as universal isometries. In the process, we get a family of homomorphisms of the affine Weyl group associated to $tilde{A}_{n-1}$ to that of $tilde{A}_{dn-1}$, indexed by natural numbers $d$, which are closed under composition.
Let $k$ be a field finitely generated over the finite field $mathbb F_p$ of odd characteristic $p$. For any K3 surface $X$ over $k$ we prove that the prime to $p$ component of the cokernel of the natural map $Br(k)to Br(X)$ is finite.
Let $V$ be a $6$-dimensional complex vector space with an involution $sigma$ of trace $0$, and let $W subseteq operatorname{Sym}^2 V^vee$ be a generic $3$-dimensional subspace of $sigma$-invariant quadratic forms. To these data we can associate an Enriques surface as the $sigma$-quotient of the complete intersection of the quadratic forms in $W$. We exhibit noncommutative Deligne-Mumford stacks together with Brauer classes whose derived categories are equivalent to those of the Enriques surfaces.
We compute explicit rational models for some Hilbert modular surfaces corresponding to square discriminants, by connecting them to moduli spaces of elliptic K3 surfaces. Since they parametrize decomposable principally polarized abelian surfaces, they are also moduli spaces for genus-2 curves covering elliptic curves via a map of fixed degree. We thereby extend classical work of Jacobi, Hermite, Bolza etc., and more recent work of Kuhn, Frey, Kani, Shaska, Volklein, Magaard and others, producing explicit families of reducible Jacobians. In particular, we produce a birational model for the moduli space of pairs (C,E) of a genus 2 curve C and elliptic curve E with a map of degree n from C to E, as well as a tautological family over the base, for 2 <= n <= 11. We also analyze the resulting models from the point of view of arithmetic geometry, and produce several interesting curves on them.
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