On Fujita's conjecture for a general hyperk\"ahler manifold in the standard series of examples

TL;DR

This paper investigates the conditions under which certain polarizations on moduli spaces of hyperk"ahler manifolds are base point free or very ample, providing new criteria and characterizations for these geometric properties.

Contribution

It introduces new criteria for base point freeness and very ampleness of polarizations on moduli spaces of hyperk"ahler manifolds of specific types, expanding understanding of their geometric properties.

Findings

Identifies cases where moduli spaces are connected.

Provides a formula for non-emptiness of these moduli spaces.

Characterizes when polarizations are base point free or very ample.

Abstract

For the moduli spaces $\Sigma_{d,t}^n$ and $\Upsilon_{d,t}^n$ of polarized hyperk\"ahler manifolds of Hilb$^n$(K3)-type and Kum$^n$-type respectively, with polarization with square $2d$ and divisibility $t$, we study general base point freeness and very ampleness of the polarization. We provide cases where these moduli space are connected and a formula characterizing when these spaces are non-empty.