Related papers: Rational curves and points on K3 surfaces

We show that if over some number field there exists a certain diagonal plane cubic curve that is locally solvable everywhere, but that does not have points over any cubic galois extension of the number field, then the algebraic part of the…

We study K3 surfaces with 9 cusps, i.e. 9 disjoint $A_2$ configurations of smooth rational curves, over algebraically closed fields of characteristic $p\neq 3$. Much like in the complex situation studied by Barth, we prove that each such…

We construct explicit examples of $K3$ surfaces over ${\mathbb Q}$ having real multiplication. Our examples are of geometric Picard rank 16. The standard method for the computation of the Picard rank provably fails for the surfaces…

The purpose of this short note is to study dominant rational maps from punctual Hilbert schemes of length $k>1$ of projective K3 surfaces $S$ containing infinitely many rational curves. Precisely, we prove that their image is necessarily…

In this paper we study the automorphisms group of some K3 surfaces which are double covers of the projective plane ramified over a smooth sextic plane curve. More precisely, we study some particlar case of a K3 surface of Picard rank two.

In this paper we compute upper bounds for the number of ordinary triple points on a hypersurface in $P^3$ and give a complete classification for degree six (degree four or less is trivial, and five is elementary). But the real purpose is to…

We study the family of rational curves on arbitrary smooth hypersurfaces of low degree using tools from analytic number theory.

In this paper, we initiate the systematic study of density of algebraic points on surfaces. We give an effective asymptotic range in which the density degree set has regular behavior dictated by the index. By contrast, in small degree, the…

This is a survey of the geometry of complex cubic fourfolds with a view toward rationality questions. Topics include classical constructions of rational examples, Hodge structures and special cubic fourfolds, associated K3 surfaces and…

We proved that every rational curves in the primitive class of a general K3 surface of any genus is nodal.

These notes will give an introduction to the theory of K3 surfaces. We begin with some general results on K3 surfaces, including the construction of their moduli space and some of its properties. We then move on to focus on the theory of…

We exhibit automorphisms of a certain K3 surface in $\mathbb{P}^1\times \mathbb{P}^1 \times \mathbb{P}^1$ with an isolated fixed point at which the induced action on the stalk of the structure sheaf is arbitrarily close to the identity.…

The aim of this paper is to describe algebraic K3 surfaces with an even set of rational curves or of nodes. Their minimal possible Picard number is nine. We completely classify these K3 surfaces and after a carefull analysis of the divisors…

In this paper we give a complete characterization of the intersections between the Norm-Trace curve over $\mathbb{F}_{q^3}$ and the curves of the form $y=ax^3+bx^2+cx+d$, generalizing a previous result by Bonini and Sala, providing more…

We study the moduli space of Hessian K3 surfaces as arithmetic quotients.

We construct an explicit, multiplicative Chow-K\"unneth decomposition for the Hilbert scheme of points of a K3 surface. We further refine this decomposition with respect to the action of the Looijenga-Lunts-Verbitsky Lie algebra.

The notion of constant cycle curves on K3 surfaces is introduced. These are curves that do not contribute to the Chow group of the ambient K3 surface. Rational curves are the most prominent examples. We show that constant cycle curves…

We study the inertia groups of some smooth rational curves on 2-elementary K3 surfaces and singular K3 surfaces from the view of topological entropy, with an application to a long standing open question of Coble on the inertia group of a…

We show how to construct non-isotrivial families of supersingular K3 surfaces over rational curves using a relative form of the Artin-Tate isomorphism and twisted analogues of Bridgeland's results on moduli spaces of stable sheaves on…

Let $X$ be a K3 surface defined over a number field $K$. Assume that $X$ admits a structure of an elliptic fibration or an infinite group of automorphisms. Then there exists a finite extension $K'/K$ such that the set of $K'$-rational…