A Compactification of the Space of Algebraic Maps from P^1 to a Grassmannian
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Author
Shao, YijunIssue Date
2010Advisor
Hu, YiCommittee Chair
Hu, Yi
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The University of Arizona.Rights
Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.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.Type
textElectronic Dissertation
Degree Name
Ph.D.Degree Level
doctoralDegree Program
MathematicsGraduate College