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Compactified Jacobians of Extended ADE Curves and Lagrangian Fibrations

Adam Czapliński, Andreas Krug, Manfred Lehn, Sönke Rollenske

Abstract

We observe that general reducible curves in sufficiently positive linear systems on K3 surfaces are of a form that generalises Kodaira's classification of singular elliptic fibres and thus call them extended ADE curves. On such a curve $C$, we describe a compactified Jacobian and show that its components reflect the intersection graph of $C$. This extends known results when $C$ is reduced, but new difficulties arise when $C$ is non-reduced. As an application, we get an explicit description of general singular fibres of certain Lagrangian fibrations of Beauville-Mukai type.

Compactified Jacobians of Extended ADE Curves and Lagrangian Fibrations

Abstract

We observe that general reducible curves in sufficiently positive linear systems on K3 surfaces are of a form that generalises Kodaira's classification of singular elliptic fibres and thus call them extended ADE curves. On such a curve , we describe a compactified Jacobian and show that its components reflect the intersection graph of . This extends known results when is reduced, but new difficulties arise when is non-reduced. As an application, we get an explicit description of general singular fibres of certain Lagrangian fibrations of Beauville-Mukai type.
Paper Structure (26 sections, 46 theorems, 148 equations, 1 figure)

This paper contains 26 sections, 46 theorems, 148 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Generalities concerning sheaves on curves
  3. Picard varieties of curves
  4. Duals of ideal sheaves of singular points
  5. Pure sheaves on divisorial curves
  6. Classification of stable sheaves on an extended ADE curve
  7. Set-up and Notation
  8. Sheaves on multiple components
  9. Line bundles on extended ADE curves
  10. Definition of Stability
  11. Condition on the polarisation
  12. Classification of stable sheaves by their components
  13. Presentation of stable sheaves as generalised divisors
  14. Uniqueness for singular stable sheaves
  15. Description of the moduli space
  16. ...and 11 more sections

Key Result

Theorem 1.1

Let $H$ be a polarisation as in ass:H. Then every semi-stable sheaf in the compactified Jacobian $\mathcal{M}_\chi(C)$ is already stable and of the form where ${\mathcal{I}}\subset {\mathcal{O}}_C$ is the ideal sheaf of a closed point $x$ and $L$ is a line bundle in a fixed component of the Picard group, the component depending on $H$ and $\chi$. The sheaf $L(x)$ is a line bundle if and only if $

Figures (1)

  • Figure 1: The extended Dynkin graphs

Theorems & Definitions (103)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1
  • proof
  • Corollary 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • ...and 93 more