Some birational geometric aspects of moduli spaces of sheaves on surfaces via bridgeland wall-crossing

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Publication Type dissertation
School or College College of Science
Department Mathematics
Author Martinez, Cristian
Title Some birational geometric aspects of moduli spaces of sheaves on surfaces via bridgeland wall-crossing
Date 2015-05
Description We study some birational geometric aspects of moduli spaces of semistable sheaves on surfaces. We observe that moduli spaces of semistable sheaves on a Del Pezzo surface are Mori-dream spaces and, following the techniques intro- duced by Arcara, Bertram, Coskun, and Huizenga, relate the Mori chambers to chambers in the Stability manifold introduced by Bridgeland. In the special case when X = P2, the wall-crossing phenomena for stability conditions (and therefore the wall-crossing in NE(X)) can be analyzed very closely. In fact, for the case of 1- dimensional sheaves with vanishing Euler-Poincare characteristic, we can describe several of the birational models, including the minimal. As we will study later, the wall-crossing phenomena in this case gives information about existence of flips of Secant varieties of Veronese surfaces. To do this we will need a generalization of a duality result of Maican for moduli spaces of 1-dimensional plane sheaves to any moduli space of Bridgeland semistable objects. In particular, we prove that Maican's result holds on arbitrary smooth complex surfaces. Finally, we show that the change of polarization for moduli spaces of sheaves on a smooth projective complex surface X, and the birational geometry of X itself, are a consequence of the wall-crossing phenomena on Stab(X).
Type Text
Publisher University of Utah
Subject Algebraic geometry; Birational geometry; Derived categories; Stability conditions
Dissertation Institution University of Utah
Dissertation Name Doctor of Philosophy
Language eng
Rights Management Copyright © Cristian Martinez 2015
Format Medium application/pdf
Format Extent 1,335,895 Bytes
Identifier etd3/id/3761
ARK ark:/87278/s6100bj4
Setname ir_etd
Date Created 2016-02-29
Date Modified 2021-05-06
ID 197312
Reference URL https://collections.lib.utah.edu/ark:/87278/s6100bj4