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

Persistent Link:
http://hdl.handle.net/10150/194715
Title:
A Compactification of the Space of Algebraic Maps from P^1 to a Grassmannian
Author:
Shao, Yijun
Issue Date:
2010
Publisher:
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:
text; Electronic Dissertation
Keywords:
blow up; moduli space; normal crossing divisor; Quot scheme
Degree Name:
Ph.D.
Degree Level:
doctoral
Degree Program:
Mathematics; Graduate College
Degree Grantor:
University of Arizona
Advisor:
Hu, Yi
Committee Chair:
Hu, Yi

Full metadata record

DC FieldValue Language
dc.language.isoenen_US
dc.titleA Compactification of the Space of Algebraic Maps from P^1 to a Grassmannianen_US
dc.creatorShao, Yijunen_US
dc.contributor.authorShao, Yijunen_US
dc.date.issued2010en_US
dc.publisherThe University of Arizona.en_US
dc.rightsCopyright © 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.en_US
dc.description.abstractLet 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.en_US
dc.typetexten_US
dc.typeElectronic Dissertationen_US
dc.subjectblow upen_US
dc.subjectmoduli spaceen_US
dc.subjectnormal crossing divisoren_US
dc.subjectQuot schemeen_US
thesis.degree.namePh.D.en_US
thesis.degree.leveldoctoralen_US
thesis.degree.disciplineMathematicsen_US
thesis.degree.disciplineGraduate Collegeen_US
thesis.degree.grantorUniversity of Arizonaen_US
dc.contributor.advisorHu, Yien_US
dc.contributor.chairHu, Yien_US
dc.contributor.committeememberCastravet, Ana-Mariaen_US
dc.contributor.committeememberJoshi, Kirtien_US
dc.contributor.committeememberThakur, Dineshen_US
dc.identifier.proquest11250en_US
dc.identifier.oclc752261094en_US
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