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Counting elliptic fibrations on K3 surfaces

Dino Festi, Davide Cesare Veniani

Abstract

We solve the problem of counting jacobian elliptic fibrations on an arbitrary complex projective K3 surface up to automorphisms. We then illustrate our method with several explicit examples.

Counting elliptic fibrations on K3 surfaces

Abstract

We solve the problem of counting jacobian elliptic fibrations on an arbitrary complex projective K3 surface up to automorphisms. We then illustrate our method with several explicit examples.

Paper Structure

This paper contains 13 sections, 31 theorems, 125 equations, 7 tables.

Table of Contents

  1. Introduction
  2. Main theorem
  3. Lattices
  4. Preliminary results
  5. Statement and corollaries
  6. Proof of \ref{['thm:main_multiplicity']}
  7. Guidelines
  8. Examples
  9. Barth--Peters family
  10. Kummer surfaces associated to a product of elliptic curves
  11. Jacobian Kummer surfaces
  12. Double covers of the plane ramified over six lines
  13. Apéry--Fermi pencil

Key Result

lemma 1

The map defined by sending a geometric embedding $\iota \colon \bU \hookrightarrow S_X$ to the fibration induced by the elliptic curve $E \coloneqq \iota(e)$ is a bijection.

Theorems & Definitions (45)

  • definition 1
  • lemma 1
  • lemma 2
  • proposition 1
  • definition 2
  • corollary 1
  • proposition 2
  • theorem 1
  • remark 1
  • corollary 2
  • ...and 35 more