Vol. 10, No. 6, 2016

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Geometry and stability of tautological bundles on Hilbert schemes of points

David Stapleton

Vol. 10 (2016), No. 6, 1173–1190
Abstract

We explore the geometry and establish the slope-stability of tautological vector bundles on Hilbert schemes of points on smooth surfaces. By establishing stability in general, we complete a series of results of Schlickewei and Wandel, who proved the slope-stability of these vector bundles for Hilbert schemes of 2 points or 3 points on K3 or abelian surfaces with Picard group restrictions. In exploring the geometry, we show that every sufficiently positive semistable vector bundle on a smooth curve arises as the restriction of a tautological vector bundle on the Hilbert scheme of points on the projective plane. Moreover, we show that the tautological bundle of the tangent bundle is naturally isomorphic to the log tangent sheaf of the exceptional divisor of the Hilbert–Chow morphism.

Keywords
Hilbert schemes of surfaces, vector bundles on surfaces, Fourier–Mukai transforms, slope-stability, spectral curves, log tangent bundle, tautological bundles, Hilbert schemes of points
Mathematical Subject Classification 2010
Primary: 14J60
Milestones
Received: 28 June 2015
Revised: 28 April 2016
Accepted: 28 May 2016
Published: 30 August 2016
Authors
David Stapleton
Department of Mathematics
Stony Brook University
Math Tower 2118
Stony Brook, NY 11794
United States