Abstract :

Given a collection of geometric objects, a fundamental question is to classify them in some manner. One way to deal with this question is by constructing and studying the geometry of Moduli spaces, which serve as parameter spaces of these objects. In this lecture, I will talk about a specific form of Moduli space called Quot Schemes which parametrize quotients of a fixed vector bundle. These spaces can also be thought of as compactifications of spaces of maps from curves to Projective spaces (and more generally Grassmanians), and they play an extremely important role in enumerative problems related to curves in Grassmanians. I will review various basic properties of the Quot scheme like its irreducibility and smoothness. In the later part of the talk, I will discuss a study of line bundles on Quot schemes and indicate how such a study can potentially lead us towards an understanding of the birational geometry of the Quot scheme. This part is a joint work with Ronnie Sebastian.