Bridgeland stability on threefolds: Some wall crossings

Schmidt, Benjamin

doi:10.1090/jag/752

Author: Benjamin Schmidt
Journal: J. Algebraic Geom. 29 (2020), 247-283
DOI: https://doi.org/10.1090/jag/752
Published electronically: November 15, 2019
MathSciNet review: 4069650
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Abstract | References | Additional Information

Abstract: Following up on the construction of Bridgeland stability condition on $\mathbb {P}^3$ by Macrì, we develop techniques to study concrete wall crossing behavior for the first time on a threefold. In some cases, such as complete intersections of two hypersurfaces of the same degree or twisted cubics, we show that there are two chambers in the stability manifold where the moduli space is given by a smooth projective irreducible variety, respectively, the Hilbert scheme. In the case of twisted cubics, we compute all walls and moduli spaces on a path between those two chambers. This allows us to give a new proof of the global structure of the main component, originally due to Ellingsrud, Piene, and Strømme. In between slope stability and Bridgeland stability there is the notion of tilt stability that is defined similarly to Bridgeland stability on surfaces. Beyond just $\mathbb {P}^3$, we develop tools to use computations in tilt stability to compute wall crossings in Bridgeland stability.

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Additional Information

Benjamin Schmidt
Affiliation: Department of Mathematics, The University of Texas at Austin, 2515 Speedway, Austin, Texas 78712
Address at time of publication: Gottfried Wilhelm Leibniz Universität Hannover, Institut für Algebraische Geometrie, Welfengarten 1, 30167 Hannover, Germany
MR Author ID: 1079080
Email: bschmidt@math.uni-hannover.de

Received by editor(s): September 19, 2016
Received by editor(s) in revised form: July 20, 2018
Published electronically: November 15, 2019
Additional Notes: This research was partially supported by NSF grants DMS-1160466 and DMS-1523496 (PI Emanuele Macrì) and a presidential fellowship of the Ohio State University.
Article copyright: © Copyright 2019 University Press, Inc.