Some families of big and stable bundles on $K3$ surfaces and on their Hilbert schemes of points

TL;DR

This paper studies the stability and bigness of vector bundles on general polarized K3 surfaces and their Hilbert schemes, introducing new conditions and results for tangent and tautological bundles, with explicit cases for small k.

Contribution

It provides new criteria for the stability and bigness of tangent and tautological bundles on K3 surfaces and their Hilbert schemes, including explicit results for small k.

Findings

Conditions for the bigness and stability of twisted tangent bundles on K3 surfaces.

Results on global generation, bigness, and stability of tautological bundles on Hilbert schemes.

Explicit analysis and computations for cases k=2,3.

Abstract

Here we investigate meaningful families of vector bundles on a very general polarized $K3$ surface $(X,H)$ and on the corresponding Hyper--Kaehler variety given by the Hilbert scheme of points $X^{[k]}:= {\rm Hilb}^k(X)$, for any integer $k \geqslant 2$. In particular, we prove results concerning bigness and stability of such bundles. First, we give conditions on integers $n$ such that the twist of the tangent bundle of $X$ by the line bundle $nH$ is big and stable on~$X$; we then prove a similar result for a natural twist of the tangent bundle of $X^{[k]}$. Next, we prove global generation, bigness and stability results for tautological bundles on $X^{[k]}$ arising either from line bundles or from Mukai-Lazarsfeld bundles, as well as from Ulrich bundles on $X$, using a careful analysis on Segre classes and numerical computations for $k = 2, 3$.