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

Dragos Oprea

Department of Mathematics, University of California San Diego, 9500 Gilman Drive, La Jolla, CA, USA

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, Boissière 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 $K3$s and minimal surfaces of general type, as well as to the punctual Quot schemes of curves.

Keywords: Hilbert scheme, Quot scheme, tautological bundles.

MSC: 14C05, 14D20, 14C17

Received: January 31, 2022; in final form July 31, 2022; Published online August 12, 2022

Language: English

DOI: 10.3842/SIGMA.2022.061

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