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class of rational surfaces with a non-rational singularity explicitly given by a single equation

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Date Issued:
2013
Summary:
The family of algebraic surfaces X dened by the single equation zn = (y a1x) (y anx)(x 1) over an algebraically closed eld k of characteristic zero, where a1; : : : ; an 2 k are distinct, is studied. It is shown that this is a rational surface with a non-rational singularity at the origin. The ideal class group of the surface is computed. The terms of the Chase-Harrison-Rosenberg seven term exact sequence on the open complement of the ramication locus of X ! A2 are computed; the Brauer group is also studied in this unramied setting. The analysis is extended to the surface eX obtained by blowing up X at the origin. The interplay between properties of eX , determined in part by the exceptional curve E lying over the origin, and the properties of X is explored. In particular, the implications that these properties have on the Picard group of the surface X are studied.
Title: A class of rational surfaces with a non-rational singularity explicitly given by a single equation.
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Name(s): Harmon, Drake.
Charles E. Schmidt College of Science
Department of Mathematical Sciences
Type of Resource: text
Genre: Electronic Thesis Or Dissertation
Date Issued: 2013
Publisher: Florida Atlantic University
Physical Form: electronic
Extent: viii, 75 p. : ill.
Language(s): English
Summary: The family of algebraic surfaces X dened by the single equation zn = (y a1x) (y anx)(x 1) over an algebraically closed eld k of characteristic zero, where a1; : : : ; an 2 k are distinct, is studied. It is shown that this is a rational surface with a non-rational singularity at the origin. The ideal class group of the surface is computed. The terms of the Chase-Harrison-Rosenberg seven term exact sequence on the open complement of the ramication locus of X ! A2 are computed; the Brauer group is also studied in this unramied setting. The analysis is extended to the surface eX obtained by blowing up X at the origin. The interplay between properties of eX , determined in part by the exceptional curve E lying over the origin, and the properties of X is explored. In particular, the implications that these properties have on the Picard group of the surface X are studied.
Identifier: 851066719 (oclc), 3360782 (digitool), FADT3360782 (IID), fau:4094 (fedora)
Note(s): by Drake Harmon.
Vita.
Thesis (Ph.D.)--Florida Atlantic University, 2013.
Includes bibliography.
Mode of access: World Wide Web.
System requirements: Adobe Reader.
Subject(s): Mathematics
Galois modules (Algebra)
Class field theory
Algebraic varieties
Integral equations
Persistent Link to This Record: http://purl.flvc.org/fcla/dt/3360782
Use and Reproduction: http://rightsstatements.org/vocab/InC/1.0/
Owner Institution: FAU