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Enumerative geometry of $K3$ surfaces

Thomas Dedieu

Abstract

The aim of these notes is to explain various enumerative results about $K3$ surfaces without assuming familiarity with Gromov--Witten theory. The enumerative results in question are due to Beauville, Bryan and Leung, Pandharipande, Maulik, Thomas, and others, and confirm conjectures made by Yau--Zaslow, Göttsche, and Katz--Klemm--Vafa.

Enumerative geometry of $K3$ surfaces

Abstract

The aim of these notes is to explain various enumerative results about surfaces without assuming familiarity with Gromov--Witten theory. The enumerative results in question are due to Beauville, Bryan and Leung, Pandharipande, Maulik, Thomas, and others, and confirm conjectures made by Yau--Zaslow, Göttsche, and Katz--Klemm--Vafa.
Paper Structure (67 sections, 22 theorems, 113 equations, 2 tables)

This paper contains 67 sections, 22 theorems, 113 equations, 2 tables.

Table of Contents

  1. Enumerative geometry of $K3$ surfaces
  2. Introduction
  3. Contents description
  4. Gromov--Witten theory
  5. Terminology and conventions
  6. Acknowledgements
  7. Rational curves in a primitive class
  8. An elementary topological counting formula
  9. Euler number
  10. Application to $K3$ surfaces of genus one
  11. Proof of the Beauville--Yau--Zaslow formula
  12. Compactified Jacobian of an integral curve
  13. Remark
  14. Remark
  15. Two fundamental interpretations of the multiplicity
  16. ...and 52 more sections

Key Result

Theorem (2.6)

Let $(S,L)$ be a smooth primitive $K3$ surface of genus $p_0$, and assume that $\mathrm{Pic} (S) = \mathbf{Z} \, L$. Then there is a finite number $N ^{p_0}$ of rational curves in the complete linear system $|L|$, and it is determined by the formula

Theorems & Definitions (36)

  • Theorem (2.6): Yau--Zaslow, Beauville
  • Lemma (2.8)
  • proof
  • Lemma (2.10)
  • Claim (2.13)
  • proof
  • Claim (2.14)
  • proof
  • Proposition (2.15)
  • proof
  • ...and 26 more