Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The 2-divisibility of divisors on K3 surfaces in characteristic 2

Toshiyuki Katsura, Shigeyuki Kondō, Matthias Schütt

Abstract

We show that K3 surfaces in characteristic 2 can admit sets of $n$ disjoint smooth rational curves whose sum is divisible by 2 in the Picard group, for each $n=8,12,16,20$. More precisely, all values occur on supersingular K3 surfaces, with exceptions only at Artin invariants 1 and 10, while on K3 surfaces of finite height, only $n=8$ is possible.

The 2-divisibility of divisors on K3 surfaces in characteristic 2

Abstract

We show that K3 surfaces in characteristic 2 can admit sets of disjoint smooth rational curves whose sum is divisible by 2 in the Picard group, for each . More precisely, all values occur on supersingular K3 surfaces, with exceptions only at Artin invariants 1 and 10, while on K3 surfaces of finite height, only is possible.

Paper Structure

This paper contains 16 sections, 37 theorems, 110 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Dualizing sheaf
  3. $(-2)$-curves on K3 surfaces
  4. Examples
  5. Case $A = \emptyset$
  6. Case $A \neq \emptyset$
  7. Case $A \neq \emptyset$ with $n = 20$
  8. The case that $\phi" : X \longrightarrow {\bf P}^1$ has a 2-section $s\neq \xi$
  9. The case that $\phi" : X \longrightarrow {\bf P}^1$ has a section $s$
  10. Concrete results for $n = 20$
  11. Quasi-elliptic fibrations with section
  12. Artin invariants $\sigma=3,4,\hdots,9$
  13. Artin invariant $\sigma=2$
  14. Artin invariant $\sigma=1$
  15. Exemplary Hierarchy of quasi-elliptic fibrations
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $X$ be a K3 surface in characteristic $p=2$. Let $n\in {\bf N}$ and assume that $X$ contains disjoint $(-2)$-curves $E_1,\hdots,E_n$ such that $\sum_{i=1}^{n}E_i$ is divisible by 2 in ${\rm Pic}(X)$. Then $n\in\{8,12,16,20\}$. More precisely,

Theorems & Definitions (81)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Corollary 3.3
  • ...and 71 more