A Compactification of the Space of Algebraic Maps from P^1 to a Grassmannian

Abstract

Let Md be the moduli space of algebraic maps (morphisms) of degree d from P^1 to a fixed Grassmannian. The main purpose of this thesis is to provide an explicit construction of a compactification of Md satisfying the following property: the compactification is a smooth projective variety and the boundary is a simple normal crossing divisor. The main tool of the construction is blowing-up. We start with a smooth compactification given by Quot scheme, which we denote by Qd. The boundary Qd\Md is singular and of high codimension. Next, we give a filtration of the boundary Qd\Md by closed subschemes: Zd,0 subset Zd,1 subset ... Zd,d-1=Qd\Md. Then we blow up the Quot scheme Qd along these subschemes succesively, and prove that the final outcome is a compactification satisfying the desired properties. The proof is based on the key observation that each Zd,r has a smooth projective variety which maps birationally onto it. This smooth projective variety, denoted by Qd,r, is a relative Quot scheme over the Quot-scheme compactification Qr for Mr. The map from Qd,r to Zd,r is an isomorphism when restricted to the preimage of Zd,r\ Zd,r-1. With the help of the Qd,r's, one can show that the final outcome of the successive blowing-up is a smooth compactification whose boundary is a simple normal crossing divisor.