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

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]:=Hilbk(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 turns out to be big and stable on X; we then prove a similar result for a natural twist of the tangent bundle of X[k]. Next, by a careful analysis on Segre classes, we prove 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.