Related papers: The maximum number of lines lying on a K3 quartic …
We prove that if $S$ is a smooth reflexive surface in $\mathbb{P}^3$ defined over a finite field $\mathbb{F}_q$, then there exists an $\mathbb{F}_q$-line meeting $S$ transversely provided that $q\geq c\operatorname{deg}(S)$, where…
K3 polytopes appear in complements of tropical quartic surfaces. They are dual to regular unimodular central triangulations of reflexive polytopes in the fourth dilation of the standard tetrahedron. Exploring these combinatorial objects, we…
We prove that the dimension of a quartic symmetroid singular along a quadric of codimension 1 is at most 4, if it is not a cone. In the maximal case, the quadric is reducible and consists of rank-3-points. If the quadric is irreducible, it…
A graph on $n \ge 3$ vertices drawn in the plane such that each edge is crossed at most four times has at most $6(n-2)$ edges -- this result proven by Ackerman is outstanding in the literature of beyond-planar graphs with regard to its…
Given a finite grid in $\mathbb{R}^2$, how many lines are needed to cover all but one point at least $k$ times? Problems of this nature have been studied for decades, with a general lower bound having been established by Ball and Serra. We…
We describe some theoretical results on triangulations of surfaces and we develop a theory on roots, decompositions and genus-surfaces. We apply this theory to describe an algorithm to list all triangulations of closed surfaces with at most…
In each characteristic $p\neq 2, 3$, it was shown in a previous work that the order of an automorphism of a K3 surface is bounded by 66, if finite. Here, it is shown that in each characteristic $p\neq 2, 3$ a K3 surface with a cyclic action…
We consider smooth surfaces $S \subset \Pq$ containing a plane curve $P$ and prove some general result concerning the linear system $|H-P|$. We then look at regular surfaces lying on hypersurfaces of degree $s$ having a plane of…
Examples of algebraic surfaces of general type with maximal Picard number are not abundant in the literature. Moreover, most known examples either possess low invariants, lie near the Noether line $K^2=2\chi-6$ or are somewhat scattered. A…
A conjecture of Serre concerns the number of rational points of bounded height on a finite cover of projective space P^{n-1}. In this paper, we achieve Serre's conjecture in the special case of smooth cyclic covers of any degree when n is…
In this paper we show that the maximum number of rational points possible for a smooth, projective, absolutely irreducible genus 4 curve over a finite field F_7 is 24. It is known that a genus 4 curve over F_7 can have at most 25 points. In…
We searched up to height $10^7$ for rational points on diagonal quartic surfaces. The computations fill several gaps in earlier lists computed by Pinch, Swinnerton-Dyer, and Bright.
Let U denote the open subset formed by deleting the unique line from the singular cubic surface x_1x_2^2+x_2x_0^2+x_3^3=0. In this paper an asymptotic formula is obtained for the number of rational points on U of bounded height, which…
We obtain several new results contributing to the theory of real equiangular line systems. Among other things, we present a new general lower bound on the maximum number of equiangular lines in d dimensional Euclidean space; we describe the…
We prove that supersingular K3 surfaces over algebraically closed fields of characteristic at least $5$ are unirational, following a simplified form of Liedtke's strategy.
We report on our project to construct non-singular cubic surfaces over $\bbQ$ with a rational line. Our method is to start with degree 4 Del Pezzo surfaces in diagonal form. For these, we develop an explicit version of Galois descent.
We determine and list all possible configurations of singular fibres on rational elliptic surfaces in characteristic three. In total, we find that 267 distinct configurations exist. This result complements Miranda and Persson's…
A desmic quartic surface is a birational model of the Kummer surface of the self-product of an elliptic curve. We recall the classical geometry of these surfaces and study their analogs in arbitrary characteristic. Moreover, we discuss the…
In 1991 S{\o}rensen proposed a conjecture for the maximum number of points on the intersection of a surface of degree $d$ and a non-degenerate Hermitian surface in $\PP^3(\Fqt)$. The conjecture was proven to be true by Edoukou in the case…
Let $\mathcal{F}$ be a plane singular curve defined over a finite field $\mathbb{F}_q$. The linear system of plane curves of a given degree passing through the singularities of $\cF$ provides potentially good bounds for the number of points…