Mukai models and Borcherds products

TL;DR

This paper links the geometry of moduli spaces of pointed K3 surfaces to orthogonal modular forms, establishing their birational properties and Kodaira dimension behavior as the number of points increases.

Contribution

It introduces an isomorphism between pluricanonical forms on moduli spaces and orthogonal modular forms, and uses Borcherds products to analyze their birational types.

Findings

Kodaira dimension stabilizes to 19 for large n

Explicit bounds for when F_{g,n} becomes nonnegative in Kodaira dimension

Identification of transition points for Kodaira dimension in certain g

Abstract

Let F_{g,n} be the moduli space of n-pointed K3 surfaces of genus g with at worst rational double points. We establish an isomorphism between the ring of pluricanonical forms on F_{g,n} and the ring of certain orthogonal modular forms, and give applications to the birational type of F_{g,n}. We prove that the Kodaira dimension of F_{g,n} stabilizes to 19 when n is sufficiently large. Then we use explicit Borcherds products to find a lower bound of n where F_{g,n} has nonnegative Kodaira dimension, and compare this with an upper bound where F_{g,n} is unirational or uniruled using Mukai models of K3 surfaces in g<21. This reveals the exact transition point of Kodaira dimension in some g.