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Derived equivalence of elliptic K3 surfaces and Jacobians

Reinder Meinsma, Evgeny Shinder

Abstract

We present a detailed study of elliptic fibrations on Fourier-Mukai partners of K3 surfaces, which we call derived elliptic structures. We fully classify derived elliptic structures in terms of Hodge-theoretic data, similar to the Derived Torelli Theorem that describes Fourier-Mukai partners. In Picard rank two, derived elliptic structures are fully determined by the Lagrangian subgroups of the discriminant group. As a consequence, we prove that for a large class of Picard rank 2 elliptic K3 surfaces all Fourier-Mukai partners are Jacobians, and we partially extend this result to non-closed fields. We also show that there exist elliptic K3 surfaces with Fourier-Mukai partners which are not Jacobians of the original K3 surface. This gives a negative answer to a question raised by Hassett and Tschinkel.

Derived equivalence of elliptic K3 surfaces and Jacobians

Abstract

We present a detailed study of elliptic fibrations on Fourier-Mukai partners of K3 surfaces, which we call derived elliptic structures. We fully classify derived elliptic structures in terms of Hodge-theoretic data, similar to the Derived Torelli Theorem that describes Fourier-Mukai partners. In Picard rank two, derived elliptic structures are fully determined by the Lagrangian subgroups of the discriminant group. As a consequence, we prove that for a large class of Picard rank 2 elliptic K3 surfaces all Fourier-Mukai partners are Jacobians, and we partially extend this result to non-closed fields. We also show that there exist elliptic K3 surfaces with Fourier-Mukai partners which are not Jacobians of the original K3 surface. This gives a negative answer to a question raised by Hassett and Tschinkel.
Paper Structure (18 sections, 46 theorems, 73 equations)

This paper contains 18 sections, 46 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Lattices
  4. K3 surfaces
  5. Căldăraru class for a non-fine moduli space
  6. Elliptic K3 surfaces
  7. Elliptic K3 surfaces of Picard rank 2
  8. Néron-Severi lattices
  9. Automorphisms and Hodge isometries
  10. Jacobians
  11. Ogg-Shafarevich Theory
  12. Isomorphisms of Jacobians
  13. $j$-special isotrivial elliptic K3 surfaces
  14. Derived equivalent K3s and Jacobians
  15. Derived elliptic structures
  16. ...and 3 more sections

Key Result

Theorem 1.2

Let $X$ be an elliptic K3 surface of Picard rank 2. Let $t$ be the multisection index of $X$ and let $2d$ be the degree of a polarisation on $X$. Denote $m=\operatorname{gcd}(d,t)$.

Theorems & Definitions (102)

  • Theorem 1.2: See Corollaries \ref{['cor:HT-DJ']} and \ref{['cor:DE-count']}
  • Proposition 1.3
  • Lemma 2.1: Nik80
  • Theorem 2.2: Surjectivity of the Period Map
  • Theorem 2.3: Torelli Theorem for K3 Surfaces
  • Theorem 2.4: Derived Torelli Theorem
  • Theorem 2.5: Counting Formula
  • Definition 2.6
  • Lemma 2.7: SZ20
  • Lemma 2.8: Cal00
  • ...and 92 more