Cones of Noether-Lefschetz divisors and moduli spaces of hyperk\"ahler manifolds

TL;DR

This paper provides a comprehensive description of the cones generated by Noether-Lefschetz divisors on orthogonal modular varieties, with applications to moduli spaces of hyperk"ahler manifolds, including uniruledness and isotriviality results.

Contribution

It introduces a general formula for the NL-cone generators and fully describes the cone and its extremal rays for specific moduli spaces, also exploring boundary divisors and positivity properties.

Findings

Explicit description of NL-cone and extremal rays for certain moduli spaces

Uniruledness of moduli spaces of hyperk"ahler manifolds of specific types

Proving isotriviality of certain polarized hyperk"ahler fourfold families

Abstract

We give a general formula for generators of the NL-cone, the cone of effective linear combinations of irreducible components of Noether-Lefschetz divisors, on an orthogonal modular variety. We then fully describe the NL-cone and its extremal rays in the cases of moduli spaces of polarized K3 surfaces and hyperk\"ahler manifolds of known deformation type for low degree polarizations. Moreover, we exhibit explicit divisors in the boundary of NL-cones for polarizations of arbitrarily large degrees. Additionally, we study the NL-positivity of the canonical class for these modular varieties. As a consequence, we obtain uniruledness results for moduli spaces of primitively polarized hyperk\"ahler manifolds of ${\rm{OG6}}$ and ${\rm{Kum}}_n$-type. Finally, we show that any family of polarized hyperk\"ahler fourfolds of ${\rm{Kum}}_2$-type with polarization of degree $2$ and divisibility $2$ over a projective base is isotrivial.