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–Kähler variety given by the Hilbert scheme of points $$X^{[k]}:= {\textrm{Hilb}}^k(X)$$ , 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 $$k = 2, 3$$ .