# Cones of Noether–Lefschetz divisors and moduli spaces of hyperkähler manifolds

## TL;DR

The paper provides a formula for generating cones in moduli spaces of hyperkähler manifolds and explores their geometric properties.

## Contribution

A general formula for generators of the Noether–Lefschetz cone on orthogonal modular varieties is derived and applied to specific moduli spaces.

## Key findings

- The NL cone of moduli spaces like polarized K3 surfaces and hyperkähler manifolds is described using minimal generators.
- Uniruledness is established for moduli spaces of primitively polarized hyperkähler manifolds of OG6 and Kum_n-type.
- Families of polarized Kum2-type manifolds with specific divisibility and polarization degree are shown to be isotrivial.

## Abstract

We give a general formula for generators of the NL cone on an orthogonal modular variety. This is the cone of effective divisors linearly equivalent to an effective linear combination of irreducible components of Noether–Lefschetz divisors. We apply this to describe, in terms of minimal generators, the NL cone of various moduli spaces of geometric origin such as those of polarized K3 surfaces, cubic fourfolds, and hyperkähler manifolds. Additionally, we establish uniruledness for many moduli spaces of primitively polarized hyperkähler manifolds of \documentclass[12pt]{minimal}
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\begin{document}$${\textrm{OG6}}$$\end{document}OG6 and \documentclass[12pt]{minimal}
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\begin{document}$${\textrm{Kum}}_n$$\end{document}Kumn-type. Finally, in analogy with the case of K3 surfaces of degree 2, we show that any family of polarized \documentclass[12pt]{minimal}
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\begin{document}$${\textrm{Kum}}_2$$\end{document}Kum2-type hyperkähler manifolds with divisibility 2 and polarization degree 2 over a projective base is isotrivial.