Related papers: K3 surfaces, rational curves, and rational points

We study the geometry of B\"uchi's K3 surface showing that the rational points of this surface are Zariski-dense.

A Nikulin configuration is the data of $16$ disjoint smooth rational curves on a K3 surface. According to results of Nikulin, the existence of a Nikulin configuration means that the K3 surface is a Kummer surface, moreover the abelian…

A conjecture of Coleman implies that only finitely many quaternion algebras over the rational numbers can be the endomorphism $\mathbf{Q}$-algebras of abelian surfaces over the complex numbers which can be defined over $\mathbf{Q}$. One may…

We prove that the number of legendrian rational cubics in $\mathbb C P^3$ through three generic points and a line is three; also we classify all legendrian curves on a quadric surface. Several computations are additionally verified using…

Let $f(z)=z^5+az^3+bz^2+cz+d \in \Z[z]$ and let us consider a del Pezzo surface of degree one given by the equation $\cal{E}_{f}: x^2-y^3-f(z)=0$. In this note we prove that if the set of rational points on the curve $E_{a,…

We consider the structure of rational points on elliptic curves in Weierstrass form. Let x(P)=A_P/B_P^2 denote the $x$-coordinate of the rational point P then we consider when B_P can be a prime power. Using Faltings' Theorem we show that…

Let $f(x)=x^5+ax^3+bx^2+cx \in \Z[x]$ and consider the hypersurface of degree five given by the equation \cal{V}_{f}: f(p)+f(q)=f(r)+f(s). Under the assumption $b\neq 0$ we show that there exists $\Q$-unirational elliptic surface contained…

We prove that, for a K3 surface in characteristic p > 2, the automorphism group acts on the nef cone with a rational polyhedral fundamental domain and on the nodal classes with finitely many orbits. As a consequence, for any non-negative…

We present a complete list of extremal elliptic K3 surfaces. There are altogether 325 of them. The first 112 coincides with Miranda-Persson's list for semi-stable ones. The data include the transcendental lattice which determines uniquely…

This paper studies curves on quartic K3 surfaces, or more generally K3 surfaces which are complete intersection in weighted projective spaces. A folklore conjecture concerning rational curves on K3 surfaces states that all K3 surfaces…

Under natural hypotheses we give an upper bound on the dimension of families of singular curves with hyperelliptic normalizations on a surface S with p_g(S) >0 via the study of the associated families of rational curves in Hilb^2(S). We use…

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…

Self-rational maps of generic algebraic K3 surfaces are conjectured to be trivial. We relate this conjecture to a conjecture concerning the irreducibility of the universal Severi varieties parametrizing nodal curves of given genus and…

We show the existence of 112 non-singular rational curves on the supersingular K3 surface with Artin invariant 1 in characteristic 3 by several ways. Using these rational curves, we have a $(16)_{10}$-configuration and a $(280_{4},…

We study the problem of determining, given an integer $k$, the rational solutions to $C_{k} : x^{3}z + x^{2} y^{2} + y^{3}z = kz^{4}$. For $k \ne 0$, the curve $C_{k}$ has genus $3$ and there are maps from $C_{k}$ to three elliptic curves…

We study complex algebraic K3 surfaces of Picard ranks 11,12, and 13 of finite automorphism group that admit a Jacobian elliptic fibration with a section of order two. We prove that the K3 surfaces admit a birational model isomorphic to a…

A rational triangle is a triangle with rational side lengths. We consider three different families of rational triangles having a fixed side and whose vertices are rational points in the plane. We display a one-to-one correspondence between…

Using a construction of Hassett--V\'arilly-Alvarado, we produce derived equivalent twisted K3 surfaces over $\mathbb{Q}$, $\mathbb{Q}_2$, and $\mathbb{R}$, where one has a rational point and the other does not. This answers negatively a…

For a real K3-surface $X$, one can introduce areas of connected components of the real point set $\mathbb{R} X$ of $X$ using a holomorphic symplectic form of $X$. These areas are defined up to simultaneous multiplication by a positive real…

Let $(X,L)$ be a polarized K3 surface of genus $g$ and $C_{en} \subset X$ be the curve of singular points of nodal elliptic curves in $|L|$. When $(X,L)$ is generic of genus two, Huybrechts observed that the curve $C_{en}$ is a constant…