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

TL;DR

This paper investigates conditions under which tautological vector bundles over Hilbert schemes of points on surfaces are big and nef, extending recent results to various surfaces including K3 and Enriques surfaces.

Contribution

It provides a simple criterion for big and nef tautological bundles on K-trivial surfaces and extends existing constructions from 2- and 3-point Hilbert schemes to arbitrary points.

Findings

Criteria for big and nef tautological bundles on K-trivial surfaces.

Extension of results from 2- and 3-point Hilbert schemes to arbitrary points.

Application of techniques to various surfaces and Quot schemes.

Abstract

We study tautological vector bundles over the Hilbert scheme of points on surfaces. For each K-trivial surface, we write down a simple criterion ensuring that the tautological bundles are big and nef, and illustrate it by examples. In the K3 case, we extend recent constructions and results of Bini, Boissi\`ere and Flamini from the Hilbert scheme of 2 and 3 points to an arbitrary number of points. Among the K-trivial surfaces, the case of Enriques surfaces is the most involved. Our techniques apply to other smooth projective surfaces, including blowups of K3s and minimal surfaces of general type, as well as to the punctual Quot schemes of curves.