Noether-Lefschetz general complete intersection K3 surfaces over the rationals

TL;DR

This paper proves that Noether-Lefschetz general polarized K3 surfaces over the rationals are Zariski dense in the moduli space for degrees up to 8, extending previous results with new techniques.

Contribution

It introduces new methods and employs Mukai's Hodge isogeny to establish Zariski density for degrees 6 and 8, expanding prior work on lower degrees.

Findings

Zariski density of Noether-Lefschetz general K3 surfaces over rationals for degrees ≤8

Extension of density results to degree 6 using novel techniques

Application of Mukai's Hodge isogeny in the proof

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.