A compactification of the space of algebraic maps from P1 to a Grassmannian

Abstract

Let M_d be the moduli space of algebraic maps (morphisms) of degree d from [special characters omitted] to a fixed Grassmannian. The main purpose of this thesis is to provide an explicit construction of a compactification of M_d 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 Q_d. The boundary Q_d \ M_d is singular and of high codimension. Next, we give a filtration of the boundary Q_d \ M_d by closed subschemes: Z_d,0 ⊂ Z _d,1 ⊂ ··· Z _d,d–1 = Q_d \ M_d. Then we blow up the Quot scheme Q_d 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 Z_d,r has a smooth projective variety which maps birationally onto it. This smooth projective variety, denoted by Q_d,r, is a relative Quot scheme over the Quot-scheme compactification Q_r for M_r. The map from Q_d,r to Z_d,r is an isomorphism when restricted to the preimage of Z_d,r \ Z_{d,r –1}. With the help of the Q_d,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.