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] := 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 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.

Big and Nef Tautological Vector Bundles over the Hilbert Scheme of Points

“…The original motivation for our work was provided by the recent results of [7], which we extend in several directions.…”

“…Compared to [7], the new ingredient is the closed form calculation of the Segre integrals in [25,26,27,33]. The formulas are explicit, and the goal here is to show how to apply them to derive geometric positivity results.…”

“…The case of K3 surfaces is the simplest and is discussed first, see Section 2. To illustrate the results of Section 2, in Section 2.1 we extend the constructions in [7] to arbitrary number of points. Other K-trivial surfaces, and in particular Enriques surfaces, are considered in Section 3.…”

“…is surjective for all zero-dimensional subschemes ζ of X of length k. Thus, via (1.1), if F is (k − 1)-very ample and k] is big and nef. This is explained for instance in [7,Propositions 2.4 and 4.5]. Let (X, H) be a polarized K3 surface, and let r = rank F .…”

“…In this note, we address the question whether the bundles F [k] → X [k] are big and nef. To our knowledge, for K3s, this has been considered for the first time in the recent article [7], alongside the stability and bigness of twists of the tangent bundle of X [k] . Specifically, if X a K3 surface of Picard rank 1, and the number of points is k = 2, 3, it is shown in [7] that F [k] is big and nef when F is either (a) a positive line bundle, (b) a twist of a Lazarsfeld-Mukai bundle (for suitable numerics), (c) a twist of an Ulrich bundle.…”