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

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 $X^{[k]}:={\mathrm{Hilb}}^k(X)$, for any integer $k⩾2$. 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 $k=2,3$.

arXiv comment21 pages. Final version, accepted for publication. The authors wish to warmly thank S. Di Rocco, A. L. Knutsen, A. F. Lopez, D. Oprea, A. Rapagnetta and A. Sarti for useful discussions and remarks, and the anonymous referee for his/her very careful reading and helpful comments