Related papers: Lines on K3 quartic surfaces in characteristic 2

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.

Cubic surfaces in characteristic two are investigated from the point of view of prime characteristic commutative algebra. In particular, we prove that, the non-Frobenius split cubic surfaces form a linear subspace of codimension four in the…

Let K be a field of characteristic different from 2 and C an elliptic curve over K given by a Weierstrass equation. To divide an element of the group C by 2, one must solve a certain quartic equation. We characterise the quartics arising…

A conic bundle or quadric bundle in characteristic 2 can have generic fiber which is nowhere smooth over the function field of the base variety. In that case, the generic fiber is called a quasilinear quadric. We solve some of the main…

The aim of these notes is to explain the remarkable formula found by Yau and Zaslow to express the number of rational curves on a K3 surface. Projective K3 surfaces fall into countably many families F(g) (g>0); a surface in F(g) admits a…

In this paper we describe six pencils of K3-surfaces which have large Picard-Number and contain precisely five singular fibers: four have A-D-E singularities and one is non-reduced. In particular we describe these surfaces as cyclic…

A cubic surface in $P^3$ is known to contain 27 lines, out of which one can form 36 Schlafli double - sixes i.e., collections $l_1,...,l_6, l'_1,..., l'_6\}$ of 12 lines such that each $l_i$ meets only $l'_j, j\neq i$ and does not meet…

In this work we classify all smooth surfaces with geometric genus equal to three and an action of a group G isomorphic to (Z/2)^k such that the quotient is a plane. We find 11 families. We compute the canonical map of all of them, finding…

In characteristic $p>0$ and for $q$ a power of $p$, we compute the number of nonplanar rational curves of arbitrary degrees on a smooth Hermitian surface of degree $q+1$ under the assumption that the curves have a parametrization given by…

We give an explicit formula for the expectation of the number of real lines on a random invariant cubic surface, i.e. a surface $Z\subset \mathbb{R}P^3$ defined by a random gaussian polynomial whose probability distribution is invariant…

Let $C \subset \mathbb{P}^3$ be a canonical curve of genus $4$ over an algebraically closed field $k$ of characteristic zero. For a line $l \subset \mathbb{P}^3$, we consider the projection $\pi_l: C \to \mathbb{P}^1$ from $l$ and the…

We classify rational, irreducible quartic symmetroids in projective 3-space. They are either singular along a line or a smooth conic section, or they have a triple point or a tacnode.

Let k=F_q be a finite field of even characteristic. We obtain in this paper a complete classification, up to k-isomorphism, of non singular quartic plane curves defined over k. We find explicit rational normal models and we give closed…

The Tate conjecture for squares of K3 surfaces over finite fields was recently proved by Ito-Ito-Koshikawa. We give a more geometric proof when the characteristic is at least 5. The main idea is to use twisted derived equivalences between…

The main theorem of "S. J. Kov\'acs: The cone of curves of a K3 surface, Math. Ann. 300 (1994), no. 4, 681-691" is proved in arbitrary characteristic. The proof is essentially the same as in the original paper where it was stated only over…

In this paper, we construct an infinite series of line arrangements in characteristic two, each featuring only triple intersection points. This finding challenges the existing conjecture that suggests the existence of only a finite number…

We study families of $K3$ surfaces obtained by double covering of the projective plane branching along curves of $(2,3)$-torus type. In the first part, we study the Picard lattices of the families, and a lattice duality of them. In the…

The correspondence between 2-parameter families of oriented lines in ${\Bbb{R}}^3$ and surfaces in $T{\Bbb{P}}^1$ is studied, and the geometric properties of the lines are related to the complex geometry of the surface. Congruences…

Let $\pi:X\rightarrow \mathbb{P}^2$ be a K3 surface of genus 2 and $L=\pi^{\ast}\mathcal{O}_{\mathbb{P}^2}(3)$, and assume that $\pi^{\ast}\mathcal{O}_{\mathbb{P}^2}(1)$ is ample as a line bundle on $X$. In this paper, we give a numerical…

By a K3-surface with nine cusps I mean a surface with nine isolated double points A_2, but otherwise smooth, such that its minimal desingularisation is a K3-surface. It is shown, that such a surface admits a cyclic triple cover branched…