Virtual Resolutions of Points in Sufficiently General Position in P^1 × P^1

Abstract

Minimal free resolutions are an important notion in algebraic geometry and commutative algebra. The minimal free resolution of a subvariety in projective spaces provides geometric properties of the subvariety. However, if the ambient space is the product of projective spaces, the minimal free resolution can be too long. On the other hand, virtual resolutions of a subvariety of products of projective spaces can be shorter and they still provide information about the subvariety. In this thesis, we investigate sets of points in P^1 × P^1 with generic Hilbert function and in particular, points in a sufficiently general position. We find an explicit virtual resolution of ideals of a sufficiently general set of points in P^1 × P^1. Our proof depends upon computing some values of the mutigraded Castelnuovo-Mumford regularity and using a result of Berkesch, Erman and Smith. We also generalize one of the Berkesch, Erman and Smith’s result in a special case.