Zero-dimensional sheaves, group actions and blowups

Abstract

The main objects of study of this thesis are 0-dimensional subschemes of affine spaces. More precisely, I have studied the following two aspects concerning them: • the interaction between 0-dimensional subschemes and linear group actions on An , • the computation of the Behrend number of 0-dimensional schemes in order to better understand the Hilbert scheme of points. In the first chapter of the thesis I have constructed the moduli spaces of certainG -equivariant coherent O A2-modules (G -constellations), introduced by Alastair Craw in 2001, which are stable with respect to a GIT stability condition. In addition, I studied the associated chamber decomposition giving an explicit combinatorial description of the chambers. In the second part of the thesis I have computed, mostly applying techniques from toric geometry, the Behrend number of a large number of fat points of the affine plane. This invariant had been abstractly defined by Behrend in 2009, but even for a scheme with only one point the (few) existing methods to calculate it could not be applied. The thesis is mainly based on the content of the following two preprints: • “Moduli spaces of Z/kZ-constellations over A2". [30, 2022] • “On the Behrend function and the blowup of some fat points", with A. T. Ricolfi.[31, 2022