Related papers: There is no Enriques surface over the integers

In this paper we study minimal algebraic surfaces with $p_g=q=1,K^2=4$ and nonhyperelliptic Albanese fibrations of genus 4. We construct for the first time a family of such surfaces as complete intersections of type $(2,3)$ in a…

We prove that there are no strictly shod algebras in hereditary gentle algebras by geometric models. As an application, we give a classification of the silted algebras for Dynkin type $\mathbb{A}_{n}$ and $\widetilde{\mathbb{A}}_{n}$.

We introduce a new class of finite dimensional gentle algebras, the surface algebras, which are constructed from an unpunctured Riemann surface with boundary and marked points by introducing cuts in internal triangles of an arbitrary…

We show that there exist genus one curves of every index over the rational numbers, answering affirmatively a question of Lang and Tate. The proof is "elementary" in the sense that it does not assume the finiteness of any Shafarevich-Tate…

We show that every classical Enriques surface containing a smooth rational curve is a Reye congruence.

We compute equations for Coughlan's family of Godeaux surfaces with torsion $\mathbb Z/2$, which we call $\mathbb Z/2$-Godeaux surfaces, and we show that it is (at most) 7 dimensional. We classify non-rational KSBA degenerations $W$ of…

Using some theory of (rational) elliptic surfaces plus elementary properties of a Mordell-Weil group regarded as module over the endomorphism ring of a (CM) elliptic curve, we present examples of such surfaces with j-invariant zero. In…

We shall study the existence condition of slope stable sheaves on Enriques surfaces. We also gives a different proof of the irreducibility of the moduli spaces of rank 2 stable sheaves.

Based on an equation for the rank of an elliptic surface over $\mathbb{Q}$ which appears in the work of Nagao, Rosen, and Silverman, we conjecture that 100% of elliptic surfaces have rank $0$ when ordered by the size of the coefficients of…

This article grew out of an effort to understand the smooth mapping class groups of certain 4-manifolds in a geometric manner. We prove a smooth analog of the Birman-Hilden theorem for manifolds that admit a hyperk\"ahler structure. This…

Given d in IN, we prove that any polarized Enriques surface (over any field of characteristic different from 2 or with a smooth K3 cover) of degree greater than 12d^2 contains at most 12 rational curves of degree at most d. For d>2 we…

Let S be a complex Enriques surface; it is the quotient of a K3 surface X by a fixed-point-free involution. The Brauer group Br(S) has a unique nonzero element. We describe its pull-back in Br(X), and show that the surfaces S for which it…

Generalizing homogeneous spectra for rings graded by natural numbers, we introduce multihomogeneous spectra for rings graded by abelian groups. Such homogeneous spectra have the same completeness properties as their classical counterparts,…

This article studies the moduli spaces of semistable objects related to two families of Enriques categories over K3 surfaces, coming from quartic double solids and special Gushel--Mukai threefolds. In particular, some classic geometric…

We prove that the product of an Enriques surface and a very general curve of genus at least 1 does not satisfy the integral Hodge conjecture for 1-cycles. This provides the first examples of smooth projective complex threefolds of Kodaira…

The simplicial wedge construction on simplicial complexes and simple polytopes has been used by a variety of authors to study toric and related spaces, including non-singular toric varieties, toric manifolds, intersections of quadrics and…

We prove an explicit surjectivity result for products of non-isotrivial, non-isogenous elliptic curves over a function field of arbitrary characteristic. This is by way of an isogeny degree bound in this setting, generated from bounds for…

A nodal Enriques surface can have at most 8 nodes. We give an explicit description of Enriques surfaces with 8 nodes, showing that they are quotients of products of elliptic curves by a group isomorphic to $\Z_2^2$ or to $\Z_2^3$ acting…

We complete the classification of smooth surfaces swept out by a 1-dimensional family of plane curves that do not form a fibration. As a consequence, we characterize manifolds swept out by a 1-dimensional family of hypersurfaces that do not…

Let $p$ be a rational prime, $q>1$ a power of $p$ and $F=\mathbb{F}_q(t)$. For an integer $d\geq 2$, let $D$ be a central division algebra over $F$ of dimension $d^2$ which is split at $\infty$ and has invariant $\mathrm{inv}_x(D)=1/d$ at…