Castelnuovo-Mumford regularity and Bridgeland stability of points in the projective plane

Abstract

In this thesis we investigate the relationship between the two invariants, Castelnuovo-Mumford regularity and Bridgeland stability of points in the pro- jective plane. The main theorem shows that for some outer Bridgeland walls, generally one invariant precisely determine the other. This means that in general the ideal sheaf destabilized along a smaller Bridgeland wall has smaller regularity than the one destabilized along a larger Bridgeland wall. In the monomial scheme case, we give a sharp inequality between the two invariants using combinatorial arguments. Finally for the general case, we prove the inequality which implies that they are somewhat close to each other.