Title page for ETD etd-0515103-151156


Type of Document Dissertation
Author Wang, Haohao
Author's Email Address hwang4@lsu.edu
URN etd-0515103-151156
Title Equations of Parametric Surfaces with Base Points via Syzygies
Degree Doctor of Philosophy (Ph.D.)
Department Mathematics
Advisory Committee
Advisor Name Title
William Adkins Committee Chair
Augusto Nobile Committee Member
Frank Neubrander Committee Member
Jerome Hoffman Committee Member
Jimmie Lawson Committee Member
Leszek Czarnecki Dean's Representative
Keywords
  • saturation
  • regularity
  • syzygy
  • base point
  • implicitization
  • biprojective space
Date of Defense 2003-05-06
Availability unrestricted
Abstract
Suppose $S$ is a parametrized surface in complex projective

3-space $mathbf{P}^3$ given as the image of $phi: mathbf{P}^1

imes mathbf{P}^1 o mathbf{P}^3$. The implicitization problem

is to compute an implicit equation $F=0$ of $S$ using the

parametrization $phi$. An algorithm using syzygies exists for

computing $F$ if $phi$ has no base points, i.e. $phi$ is

everywhere defined. This work is an extension of this algorithm to

the case of a surface with multiple base points of total

multiplicity k.

We accomplish this in three chapters. In Chapter 2, we develop

the concept and properties of Castelnuovo-Mumford regularity in

biprojective spaces. In Chapter 3, we give a criterion for

regularity in biprojective spaces. These results are applied to the

implicitization problem in Chapter 4.

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