On some moduli spaces of bundles on K3 surfaces, II

TL;DR

This paper explores conditions on polarized K3 surfaces where the moduli space of sheaves is birationally equivalent to a Hilbert scheme, extending previous results to broader cases with infinite divisorial conditions.

Contribution

It provides numerous examples of divisorial conditions on moduli spaces of polarized K3 surfaces that relate the moduli of sheaves to Hilbert schemes, generalizing earlier findings.

Findings

Existence of infinitely many divisorial conditions on moduli spaces

Birational equivalence between sheaf moduli and Hilbert schemes for general K3 surfaces

Extension of previous results to new cases with broader parameters

Abstract

We give many examples in which there exist infinitely many divisorial conditions on the moduli space of polarized K3 surfaces $(S,H)$ of degree $H^2=2g-2$, $g \geq 3$, and Picard number $rk N(S)=\rho(S)=2$ such that for a general K3 surface $S$ satisfying these conditions the moduli space of sheaves $M_S(r,H,s)$ is birationally equivalent to the Hilbert scheme $S[g-rs]$ of zero-dimensional subschemes of $S$ of lenght equal to $g-rs$. This result generalizes the main result of \cite{Nik1} when $g=rs+1$ and of \cite{Monat} when $r=s=2$, $g \geq 5$.