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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