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Kummer surfaces and quadratic line complexes in characteristic two

Toshiyuki Katsura, Shigeyuki Kondo

TL;DR

<3-5 sentence high-level summary>This paper develops a characteristic-2 analogue of the classical theory connecting quadratic line complexes, genus-2 curves, and Kummer surfaces. It introduces canonical forms for pencils of quadrics in P^5 tailored to the 2-rank of the Jacobian, and shows how ordinary, rank-1, and supersingular cases yield three interrelated geometries: Kummer quartic surfaces S,S1,S0 and their K3 resolutions via intersections of quadrics in P^5. It constructs genus-2 curves C,C1,C0 from these pencils, describes the associated Jacobians and theta divisors, and explicates the linear systems |2C| and |4C| to realize the Kummer quotients and triple-quadric intersections; the work also links to Igusa normal forms and Duquesne's results in characteristic two. Overall, the results provide a robust framework for understanding genus-2 curves and their Kummer geometries in characteristic two, enriching the classical theory with explicit equations and birational maps.

Abstract

In this paper, we study the classical theory of quadratic line complexes and Kummer surfaces. A quadratic line complex is the intersection of the Grassmannian $G(2,4)$ and a quadric hypersurface in ${\bf P}^5$, and a Kummer surface is the quotient of the Jacobian of a curve of genus 2 by the inversion. F. Klein discovered a relationship between a quadratic line complex and a curve of genus 2, its Jacobian and the associated Kummer surface. This theory holds in any characteristic not equal to two. However the situation in characteristic two is entirely different. The purpose of this paper is to give an analogue in characteristic 2 of this classical theory.

Kummer surfaces and quadratic line complexes in characteristic two

TL;DR

<3-5 sentence high-level summary>This paper develops a characteristic-2 analogue of the classical theory connecting quadratic line complexes, genus-2 curves, and Kummer surfaces. It introduces canonical forms for pencils of quadrics in P^5 tailored to the 2-rank of the Jacobian, and shows how ordinary, rank-1, and supersingular cases yield three interrelated geometries: Kummer quartic surfaces S,S1,S0 and their K3 resolutions via intersections of quadrics in P^5. It constructs genus-2 curves C,C1,C0 from these pencils, describes the associated Jacobians and theta divisors, and explicates the linear systems |2C| and |4C| to realize the Kummer quotients and triple-quadric intersections; the work also links to Igusa normal forms and Duquesne's results in characteristic two. Overall, the results provide a robust framework for understanding genus-2 curves and their Kummer geometries in characteristic two, enriching the classical theory with explicit equations and birational maps.

Abstract

In this paper, we study the classical theory of quadratic line complexes and Kummer surfaces. A quadratic line complex is the intersection of the Grassmannian and a quadric hypersurface in , and a Kummer surface is the quotient of the Jacobian of a curve of genus 2 by the inversion. F. Klein discovered a relationship between a quadratic line complex and a curve of genus 2, its Jacobian and the associated Kummer surface. This theory holds in any characteristic not equal to two. However the situation in characteristic two is entirely different. The purpose of this paper is to give an analogue in characteristic 2 of this classical theory.
Paper Structure (16 sections, 47 theorems, 267 equations, 3 figures, 7 tables)

This paper contains 16 sections, 47 theorems, 267 equations, 3 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Classical theory of Quadratic line complexes and Kummer surfaces
  4. Quadratic forms in characteristic two
  5. Pencils of quadratics in ${\bf P}^5$
  6. Kummer quartic surfaces and curves of genus 2 in characteristic two
  7. The intersection of two quadrics
  8. The quartic surfaces
  9. The intersection of three quadrics
  10. Curves of genus 2
  11. General theory
  12. Multi-linear systems
  13. Points on the theta divisor $C$
  14. The linear system $\vert 2C\vert$
  15. The linear system $\vert 4C\vert$
  16. ...and 1 more sections

Key Result

Theorem 1.1

(a) The surface $S$ is given by the equation (b) The surface $S_1$ is given by the equation (c) The surface $S_0$ is given by the equation

Figures (3)

  • Figure 1: Dual graphs of minimal resolutions of singularities
  • Figure 2: Eight lines on $\Sigma$
  • Figure 3: The dual graph of 20 $(-2)$-curves on $\widetilde{\Sigma}$

Theorems & Definitions (100)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Proposition 2.1
  • Remark 2.2
  • Proposition 2.3
  • Proposition 2.4
  • proof
  • Remark 2.5
  • ...and 90 more