Modification of Simpson moduli spaces of 1-dimensional sheaves by vector bundles, an experimental example

ii Oleksandr Iena. “Modification of Simpson moduli spaces of 1-dimensional sheaves by vector bundles, an experimental example”. Abstract. This thesis deals with the following question. Given a moduli space of coherent sheaves on a projective variety with a fixed Hilbert polynomial, to find a natural construction that replaces the subvariety of the sheaves that are not locally free on their support (we call such sheaves singular) by some variety consisting of sheaves that are locally free on their support. We consider this problem on the example of the coherent sheaves on P2 with Hilbert polynomial 3m+ 1. Given a singular coherent sheaf F with singular curve C as its support we replace F by locally free sheaves E supported on a reducible curve C0∪C1, where C0 is a partial normalization of C and C1 is an extra curve bearing the degree of E. These bundles resemble the bundles considered by Nagaraj and Seshadri (cf. [19], [20], [26]). Many properties of the singular 3m+1 sheaves are inherited by the new sheaves we introduce in this thesis (we call them R-bundles). We consider R-bundles as natural replacements of the singular sheaves