Vector bundles on an elliptic curve over a discrete valuation ring

Persistent Link:
http://hdl.handle.net/10150/280386
Title:
Vector bundles on an elliptic curve over a discrete valuation ring
Author:
Kim, Seog Young
Issue Date:
2001
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:
We classify rank 2 vector bundles on a smooth curve X of genus 1 over a discrete valuation ring R. Atiyah [5] classified rank 2 vector bundles on elliptic curves over algebraically closed fields. The fact that a genus 1 curve over a discrete valuation ring has a codimension 2 subscheme prevents us from applying Atiyah's work directly. We find that genus 1 curve over an arbitrary field can have three types of rank 2 vector bundles. We classify rank 2 vector bundles on a curve of genus 1 over a discrete valuation ring using the classification on a curve of genus 1 over a field and quadruples (L,M,Z, η) where L and M are line bundles on X and Z is a local complete intersection subscheme of codimension 2 and η is an orbit in Ext¹(M⊗I(z),L ) under the R* action.
Type:
text; Dissertation-Reproduction (electronic)
Keywords:
Mathematics.
Degree Name:
Ph.D.
Degree Level:
doctoral
Degree Program:
Graduate College; Mathematics
Degree Grantor:
University of Arizona
Advisor:
Kim, Minhyong

Full metadata record

DC FieldValue Language
dc.language.isoen_USen_US
dc.titleVector bundles on an elliptic curve over a discrete valuation ringen_US
dc.creatorKim, Seog Youngen_US
dc.contributor.authorKim, Seog Youngen_US
dc.date.issued2001en_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.abstractWe classify rank 2 vector bundles on a smooth curve X of genus 1 over a discrete valuation ring R. Atiyah [5] classified rank 2 vector bundles on elliptic curves over algebraically closed fields. The fact that a genus 1 curve over a discrete valuation ring has a codimension 2 subscheme prevents us from applying Atiyah's work directly. We find that genus 1 curve over an arbitrary field can have three types of rank 2 vector bundles. We classify rank 2 vector bundles on a curve of genus 1 over a discrete valuation ring using the classification on a curve of genus 1 over a field and quadruples (L,M,Z, η) where L and M are line bundles on X and Z is a local complete intersection subscheme of codimension 2 and η is an orbit in Ext¹(M⊗I(z),L ) under the R* action.en_US
dc.typetexten_US
dc.typeDissertation-Reproduction (electronic)en_US
dc.subjectMathematics.en_US
thesis.degree.namePh.D.en_US
thesis.degree.leveldoctoralen_US
thesis.degree.disciplineGraduate Collegeen_US
thesis.degree.disciplineMathematicsen_US
thesis.degree.grantorUniversity of Arizonaen_US
dc.contributor.advisorKim, Minhyongen_US
dc.identifier.proquest3010243en_US
dc.identifier.bibrecord.b41711154en_US
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