On the irrationality of moduli spaces of K3 surfaces

TL;DR

This paper investigates how the degrees of irrationality of moduli spaces of polarized K3 surfaces grow with genus, establishing polynomial bounds and leveraging modularity and geometric constructions.

Contribution

It provides new polynomial bounds on the irrationality degrees of K3 moduli spaces, refining previous estimates and connecting them with modular forms and special geometric cases.

Findings

Irrationality degrees grow polynomially with genus, bounded by degree 14+ε.

For certain genera, bounds are refined to degree 10 polynomials.

The proof uses modularity of Heegner divisors and special geometric constructions.

Abstract

We study how the degrees of irrationality of moduli spaces of polarized K3 surfaces grow with respect to the genus $g$. We prove that the growth is bounded by a polynomial function of degree $14+\varepsilon$ for any $\varepsilon>0$ and, for three sets of infinitely many genera, the bounds can be refined to polynomials of degree $10$. The main ingredients in our proof are the modularity of the generating series of Heegner divisors due to Borcherds and its generalization to higher codimensions due to Kudla, Millson, Zhang, Bruinier, and Westerholt-Raum. For special genera, the proof is also built upon the existence of K3 surfaces associated Hodge theoretically with certain cubic fourfolds, Gushel-Mukai fourfolds, and hyperk\"ahler fourfolds.