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Noether-Lefschetz general complete intersection K3 surfaces over the rationals

Asher Auel, Henry Scheible

Abstract

We prove that the locus of Noether-Lefschetz general polarized K3 surfaces of degree at most 8 defined over the rational numbers is Zariski dense in the moduli space. Previously, this was proved by van Luijk in the quartic case, and it follows from work of Elsenhans and Jahnel in the degree 2 case. Innovations on their methods, and employing Mukai's Hodge isogeny, suffices to handle the degree 8 case. New methods allow us to deal with the case of degree 6.

Noether-Lefschetz general complete intersection K3 surfaces over the rationals

Abstract

We prove that the locus of Noether-Lefschetz general polarized K3 surfaces of degree at most 8 defined over the rational numbers is Zariski dense in the moduli space. Previously, this was proved by van Luijk in the quartic case, and it follows from work of Elsenhans and Jahnel in the degree 2 case. Innovations on their methods, and employing Mukai's Hodge isogeny, suffices to handle the degree 8 case. New methods allow us to deal with the case of degree 6.
Paper Structure (4 sections, 15 theorems, 21 equations)

This paper contains 4 sections, 15 theorems, 21 equations.

Table of Contents

  1. Prerequisites
  2. Degree 8 Case
  3. Degree 6 Case
  4. Zariski density

Key Result

Theorem 1

The set of Noether--Lefschetz general K3 surfaces defined over $\mathbb{Q}$ is Zariski dense in $\mathcal{K}_d$ for $d \leq 8$.

Theorems & Definitions (31)

  • Theorem 1
  • Lemma 1.1: Elsenhans and Jahnel elsenhans_picard_2011
  • Theorem 1.2
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • Proposition 2.3
  • proof
  • Example 2.4
  • Corollary 2.5
  • ...and 21 more