Related papers: Normal forms for Kummer surfaces

We study projective models of generalized Kummer fourfolds via O'Grady's theta groups and the classical Coble cubic. More precisely, we establish a duality between two singular models of the generalized Kummer fourfold of a Jacobian abelian…

We study the Kodaira dimension of Kuga varieties $\mathcal{X}^n_p$ associated to the moduli spaces $\mathcal{A}_p$ of $(1, p)$-polarised abelian surfaces with level structure for prime $p \geq 3$.

We compute the class of arithmetic genus two Teichmueller curves in the Picard group of pseudo-Hilbert modular surfaces, distinguished according to their torsion order and spin invariant. As an application, we compute the number of genus…

Principally polarized abelian surfaces with prescribed real multiplication (RM) are parametrized by certain Hilbert modular surfaces. Thus rational genus 2 curves correspond to rational points on the Hilbert modular surfaces via their…

We construct smooth minimal complex surfaces of general type with $K^2=7$ and: $p_g=q=2,$ Albanese map of degree $2$ onto a $(1,2)$-polarized abelian surface; $p_g=q=1$ as a double cover of a quartic Kummer surface; $p_g=q=0$ as a double…

We introduce tropical Kummer quartic surfaces in tropical projective $3$-space as the images of certain principally polarized tropical abelian surfaces under tropical theta functions of second order. We study some of their properties,…

The Coble cubics were discovered more than a century ago in connection with genus two Riemann surfaces and theta functions. They have attracted renewed interest ever since. Recently, they were reinterpreted in terms of alternating…

We study a normalized version of the second order renormalization group flow on closed Riemannian surfaces. We discuss some general properties of this flow and establish several basic formulas. In particular, we focus on surfaces with zero…

We show that abelian surfaces (and consequently curves of genus 2) over totally real fields are potentially modular. As a consequence, we obtain the expected meromorphic continuation and functional equations of their Hasse--Weil zeta…

We provide a unique normal form for rank two irregular connections on the Riemann sphere.In fact, we provide a birational model where we introduce apparent singular points and where the bundlehas a fixed Birkhoff-Grothendieck decomposition.…

In arXiv:1710.01672, we obtained general type results for orthogonal modular varieties associated with moduli spaces of compact hyperk\"ahler manifolds of deformation generalised Kummer type (also known as 'deformation generalised Kummer…

Inspired by a theorem of Gruson-Lazarsfeld-Peskine bounding the Castelnuovo-Mumford regularity of curves in projective spaces, we bound the Theta-regularity of curves in polarized abelian varieties.

We study the equations of abelian surfaces embedded in P^{n-1} with a line bundle of polarization of type (1,n). For n>9, we show that the ideal of a general abelian surface with this polarization is generated by quadrics, and if the…

We study generalized Kummer surfaces Km$_{3}(A)$, by which we mean the K3 surfaces obtained by desingularization of the quotient of an abelian surface $A$ by an order $3$ symplectic automorphism group. Such a surface carries $9$ disjoint…

We consider a family of abelian surfaces over $\mathbb{Q}$ arising as Prym varieties of double covers of genus-$1$ curves by genus-$3$ curves. These abelian surfaces carry a polarization of type $(1,2)$ and we show that the average size of…

The classical Kummer construction attaches to an abelian surface a K3 surface. As Shioda and Katsura showed, this construction breaks down for supersingular abelian surfaces in characteristic two. Replacing supersingular abelian surfaces by…

In this article we study derived (auto)equivalences of generalized Kummer varieties $\mathrm{Kum}^n(A)$. We provide an answer to a question raised by Namikawa by showing that the generalized Kummer varieties $\mathrm{Kum}^n(A)$ and…

In this note, we calculate the boundary of movable cones and nef cones of the generalized Kummer 4-fold $\mathrm{Km}^2(A)$ attached to an abelian surface $A$ with $\mathrm{rkNS}(A) = 1$.

A generalized Kummer surface $X$ obtained as the quotient of an abelian surface by a symplectic automorphism of order 3 contains a $9\mathbf{A}_{2}$-configuration of $(-2)$-curves. Such a configuration plays the role of the…

We introduce Kummer surfaces X=Km(CxC) with the group scheme G=mu_2 acting on the self-product of the rational cuspidal curve in characteristic two. The resulting quotients are normal surfaces having a configuration of sixteen rational…