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Low degree rational curves on quasi-polarized K3 surfaces

Sławomir Rams, Matthias Schütt

Abstract

We prove that there are at most $(24-r_0)$ low-degree rational curves on high-degree models of K3 surfaces with at most Du Val singularities, where $r_0$ is the number of exceptional divisors on the minimal resolution. We also provide several existence results in the above setting (i.e. for rational curves on quasi-polarized K3 surfaces), which imply that for various values of $r_0$ our bound cannot be improved.

Low degree rational curves on quasi-polarized K3 surfaces

Abstract

We prove that there are at most low-degree rational curves on high-degree models of K3 surfaces with at most Du Val singularities, where is the number of exceptional divisors on the minimal resolution. We also provide several existence results in the above setting (i.e. for rational curves on quasi-polarized K3 surfaces), which imply that for various values of our bound cannot be improved.
Paper Structure (27 sections, 15 theorems, 75 equations)

This paper contains 27 sections, 15 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. Set-up
  3. Extended Fano graphs
  4. Proof of Theorem \ref{['thm']} (i), (ii)
  5. Proof of Theorem \ref{['thm']} (i)
  6. Proof of Theorem \ref{['thm']} (ii)
  7. Proof of Theorem \ref{['thm']} (iii)
  8. Proof of Theorem \ref{['thm']} (iii)
  9. Warm-up for the proof of Theorem \ref{['thm:c_0']}
  10. Strategy of the proof of Theorem \ref{['thm:c_0']}
  11. Proof of Theorem \ref{['thm:c_0']} in characteristic zero
  12. Proof of Theorem \ref{['thm:c_0']} in positive characteristic
  13. Proof of Theorem \ref{['thm']} (iv), (v)
  14. $p\equiv 1\mod 4, (-c_0/p)=0$ or $-1$
  15. $p\equiv 1\mod 4, (-c_0/p)=1$
  16. ...and 12 more sections

Key Result

Theorem 1.1

Let $d\in{\mathbb N}$ and let $p \neq 2,3$.

Theorems & Definitions (41)

  • Theorem 1.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Lemma 4.1
  • ...and 31 more