K3 surfaces associated to curves of genus two (original) (raw)
K3 Surfaces Associated with Curves of Genus Two
International Mathematics Research Notices, 2010
It is known ([10], [27]) that there is a unique K3 surface X which corresponds to a genus 2 curve C such that X has a Shioda-Inose structure with quotient birational to the Kummer surface of the Jacobian of C. In this paper we give an explicit realization of X as an elliptic surface over P 1 with specified singular fibers of type II * and III *. We describe how the Weierstrass coefficients are related to the Igusa-Clebsch invariants of C.
Bulletin of the London Mathematical Society, 2006
In a recent paper Ahlgren, Ono and Penniston described the L-series of K3 surfaces from a certain one-parameter family in terms of those of a particular family of elliptic curves. The Tate conjecture predicts the existence of a correspondence between these K3 surfaces and certain Kummer surfaces related to these elliptic curves. A geometric construction of this correspondence is given here, using results of D. Morrison on Nikulin involutions.
K3 surfaces and equations for Hilbert modular surfaces
Algebra & Number Theory, 2014
We outline a method to compute rational models for the Hilbert modular surfaces Y − (D), which are coarse moduli spaces for principally polarized abelian surfaces with real multiplication by the ring of integers in Q( √ D), via moduli spaces of elliptic K3 surfaces with a Shioda-Inose structure. In particular, we compute equations for all thirty fundamental discriminants D with 1 < D < 100, and analyze rational points and curves on these Hilbert modular surfaces, producing examples of genus-2 curves over Q whose Jacobians have real multiplication over Q.
Enriques surfaces and Jacobian elliptic K3 surfaces
Mathematische Zeitschrift, 2010
This paper proposes a new geometric construction of Enriques surfaces. Its starting point are K3 surfaces with jacobian elliptic fibration which arise from rational elliptic surfaces by a quadratic base change. The Enriques surfaces obtained in this way are characterised by elliptic fibrations with a rational curve as bisection which splits into two sections on the covering K3 surface. The construction has applications to the study of Enriques surfaces with specific automorphisms. It also allows us to answer a question of Beauville about Enriques surfaces whose Brauer groups show an exceptional behaviour. In a forthcoming paper, we will study arithmetic consequences of our construction.
Elliptic fibrations on a generic Jacobian Kummer surface
Journal of Algebraic Geometry, 2014
We describe all the elliptic fibrations with section on the Kummer surface X X of the Jacobian of a very general curve C C of genus 2 2 over an algebraically closed field of characteristic 0, modulo the automorphism group of X X and the symmetric group on the Weierstrass points of C C . In particular, we compute elliptic parameters and Weierstrass equations for the 25 different fibrations and analyze the reducible fibers and Mordell–Weil lattices. This answers completely a question posed by Kuwata and Shioda in 2008.
The Michigan Mathematical Journal, 2007
We determine all possible configurations of rational double points on complex normal algebraic K3 surfaces, and on normal supersingular K3 surfaces in characteristic p > 19.
Perfect points on genus one curves and consequences for supersingular K3 surfaces
arXiv: Algebraic Geometry, 2019
We describe a method to show that certain elliptic surfaces do not admit purely inseparable multisections (equivalently, that genus one curves over function fields admit no points over the perfect closure of the base field) and use it to show that any non-Jacobian elliptic structure on a very general supersingular K3 surface has no purely inseparable multisections. We also describe specific examples of such fibrations without purely inseparable multisections. Finally, we discuss the consequences for the claimed proof of the Artin conjecture on unirationality of supersingular K3 surfaces.
Perfect points on curves of genus 1 and consequences for supersingular K3 surfaces
Compositio Mathematica
We describe a method to show that certain elliptic surfaces do not admit purely inseparable multisections (equivalently, that genus 1 curves over function fields admit no points over the perfect closure of the base field) and use it to show that any non-Jacobian elliptic structure on a very general supersingular K3 surface has no purely inseparable multisections. We also describe specific examples of genus 1 fibrations on supersingular K3 surfaces without purely inseparable multisections.