Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Inseparable Kummer surfaces

Yuya Matsumoto

Abstract

We introduce an inseparable version of Kummer surfaces. It is defined as a supersingular K3 surface in characteristic 2 with 16 smooth rational curves forming a certain configuration and satisfying a suitable divisibility condition. The main result is that such a surface admits an inseparable double covering by a non-normal surface $A$ that is similar to abelian surfaces in two aspects: its numerical invariants are the same as abelian surfaces, and its smooth locus admits an abelian group structure.

Inseparable Kummer surfaces

Abstract

We introduce an inseparable version of Kummer surfaces. It is defined as a supersingular K3 surface in characteristic 2 with 16 smooth rational curves forming a certain configuration and satisfying a suitable divisibility condition. The main result is that such a surface admits an inseparable double covering by a non-normal surface that is similar to abelian surfaces in two aspects: its numerical invariants are the same as abelian surfaces, and its smooth locus admits an abelian group structure.
Paper Structure (40 sections, 51 theorems, 44 equations, 1 figure, 2 tables)

This paper contains 40 sections, 51 theorems, 44 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Main idea
  4. Results
  5. Relations with earlier works
  6. Structure of this paper
  7. Preliminaries
  8. Derivations and quotients
  9. Cartier operator on differential forms
  10. Lattices
  11. Non-taut rational double points and relation to the height of K3 surfaces
  12. Picard lattices of supersingular K3 surfaces
  13. Divisors on K3 surfaces
  14. Kummer lattices
  15. Overlattices of $A_1^{\oplus m}$
  16. ...and 25 more sections

Key Result

Proposition 2.1

Let $Y$ be a scheme over a field $k$ of characteristic $p > 0$. Moreover, the quotient by the derivation and the quotient by the group scheme action are equal and, in the case of $\mu_p$, also equal to $(\mathcal{O}_Y)_0$.

Figures (1)

  • Figure 1: Visualization of $Q_4$ and $Q_2$

Theorems & Definitions (114)

  • Proposition 2.1
  • proof
  • Theorem 2.2: Katsura--Takeda formula Katsura--Takeda:quotients*Proposition 2.1
  • Proposition 2.3: Matsumoto:k3alphap*Lemma 2.11
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • Lemma 2.7
  • ...and 104 more