Unirational moduli spaces of some elliptic K3 surfaces

TL;DR

This paper demonstrates the unirationality of moduli spaces of certain polarized K3 surfaces for specific parameters, using projective models of elliptic K3 surfaces with particular curve configurations.

Contribution

It establishes the unirationality of moduli spaces of $Uigoplus raket{-2k}$-polarized K3 surfaces for many values of $k$, expanding understanding of their geometric properties.

Findings

Unirationality proven for $k otin ext{excluded set}$ up to 97.

Systematic study of elliptic K3 surface models in projective spaces.

Identification of specific curve configurations in projective models.

Abstract

We show that the moduli space of $U\oplus \langle -2k \rangle$-polarized K3 surfaces is unirational for $k \le 50$ and $k \notin \{11,35,42,48\}$, and for other several values of $k$ up to $k=97$. Our proof is based on a systematic study of the projective models of elliptic K3 surfaces in $\mathbb{P}^n$ for $3\le n \le 5$ containing either the union of two rational curves or the union of a rational and an elliptic curve intersecting at one point.