Related papers: Integral points on conic log K3 surfaces

We develop a heuristic for the density of integer points on affine cubic surfaces. Our heuristic applies to smooth surfaces defined by cubic polynomials that are log K3, but it can also be adjusted to handle singular cubic surfaces. We…

Profitant du travail de pr\'ec\'edent d'Harpaz nous utilisons la m\'ethode de descente-fibration de Swinnerton-Dyer pour \'etudier les points int\'egraux sur des surfaces affines qui sont des fibration de tores de norme 1 sur…

In this paper we prove that the set of $S$-integral points of the smooth cubic surfaces in $\mathbb{A}^3$ over a number field $k$ is not thin, for suitable $k$ and $S$. As a corollary, we obtain results on the complement in $\mathbb{P}^2$…

We construct families of log K3 surfaces and study the arithmetic of their members. We use this to produce explicit surfaces with an order 5 Brauer-Manin obstruction to the integral Hasse principle.

For K\"ahler K3 surfaces we consider Kulikov models of type III tamed by a symplectic form. Our main result shows that the generic smooth fiber admits an almost toric fibration over the intersection complex, which inherits a natural nodal…

In this article we will show that there are infinitely many symmetric, integral 3 x 3 matrices, with zeros on the diagonal, whose eigenvalues are all integral. We will do this by proving that the rational points on a certain non-Kummer,…

Following recent works by E. Fuchs et al. and by the author, we study rational and integral points on Markoff-type K3 (MK3) surfaces, i.e., Wehler K3 surfaces of Markoff type. In particular, we construct a family of MK3 surfaces which have…

Let $Y$ be the complement of a plane quartic curve $D$ defined over a number field. Our main theorem confirms the Lang-Vojta conjecture for $Y$ when $D$ is a generic smooth quartic curve, by showing that its integral points are confined in…

We give criteria for the existence of geometric smoothings of a proper lci scheme or a DM stack $X$ as well as for a polarized lci scheme $(X,L)$, without assuming that $X$ is reduced. As applications, we give criteria for the smoothability…

We construct K3 surfaces over number fields that have good reduction everywhere. These do not exists over the rational numbers, by results of Abrashkin and Fontaine. Our surfaces exist for three quadratic number fields, and an infinite…

The aim of this paper is to generalize results known for the symplectic involutions on K3 surfaces to the order 3 symplectic automorphisms on K3 surfaces. In particular, we will explicitly describe the action induced on the lattice…

We consider threefolds that admit a fibration by K3 surfaces over a nonsingular curve, equipped with a divisorial sheaf that defines a polarisation of degree two on the general fibre. Under certain assumptions on the threefold we show that…

We give necessary and sufficient conditions for a big and nef line bundle L of any degree on a K3 surface or Enriques surface to be k-very ample and k-spanned. Furthermore, we give necessary and sufficient conditions for a spanned and big…

We prove that the locus of Hilbert schemes of n points on a projective K3 surface is dense in the moduli space of irreducible holomorphic symplectic manifolds of that deformation type. The analogous result for generalized Kummer manifolds…

This survey focuses on the geometric problem of log-surfaces, which are pairs consisting of a smooth projective surface and a reduced non-empty boundary divisor. In the first part, we focus on the geography problem for complex log-surfaces…

We study integral points on affine surfaces by means of a new method, relying on the Subspace Theorem. Under suitable assumptions on the divisor at infinity, we prove that the integral points are contained in a curve. As a corollary, we…

We characterize all projective K3 surfaces on which every integral pseudoeffective divisor admits an integral Zariski decomposition, using an explicit, terminating finite-step algorithm.

We show several examples of integrable systems related to special K3 and rational surfaces (e.g., an elliptic K3 surface, a K3 surface given by a double covering of the projective plane, a rational elliptic surface, etc.). The construction,…

We use classification of non-symplectic automorphisms of K3 surfaces to obtain a partial classification of log del Pezzo surfaces of index three. We can classify those with "Multiple Smooth Divisor Property", whose definition we will give.…

Let $X^o=\mathbb P^3\setminus D$ where $D$ is the union of two quadrics such that their intersection contains a smooth conic, or the union of a smooth quadric surface and two planes, or the union of a smooth cubic surface $V$ and a plane…