Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 061, 21 pages      arXiv:2208.06599
Contribution to the Special Issue on Enumerative and Gauge-Theoretic Invariants in honor of Lothar Göttsche on the occasion of his 60th birthday

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

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

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.

Key words: Hilbert scheme; Quot scheme; tautological bundles.

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