University of Illinois Urbana-Champaign

Syzygies and implicitization of tensor product surfaces

Duarte Gelvez, Eliana Maria

Content Files
DUARTEGELVEZ-DISSERTATION-2017.pdf

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https://hdl.handle.net/2142/97285

Description

Title
Syzygies and implicitization of tensor product surfaces
Author(s)
Duarte Gelvez, Eliana Maria
Issue Date
2017-04-10
Director of Research (if dissertation) or Advisor (if thesis)
Schenck, Henry
Doctoral Committee Chair(s)
Nevins, Thomas
Committee Member(s)
  • Francis, George
  • Reznick, Bruce
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Date of Ingest
2017-08-10T19:14:37Z
Keyword(s)
  • Implicitization
  • Syzygy
  • Rees algebras
  • Basepoints
  • Tensor product surface
  • Smooth quadric
Abstract
A tensor product surface is the closure of the image of a rational map λ : P1 ×P1-->P3. These surfaces arise in geometric modeling and in this context it is useful to know the implicit equation of λ in P3. Currently, syzygies and Rees algebras provide the fastest and most versatile method to find implicit equations of parameterized surfaces. Knowing the structure of the syzygies of the polynomials that define the map λ allows us to formulate faster algorithms for implicitization of these surfaces and also to understand their singularities. We show that for tensor product surfaces without basepoints, the existence of a linear syzygy imposes strong conditions on the structure of the syzygies that determine the implicit equation. For tensor product surfaces with basepoints we show that the syzygies that determine the implicit equation of λ are closely related to the geometry of the set of points at which λ is undefined.
Graduation Semester
2017-05
Type of Resource
text
Permalink
http://hdl.handle.net/2142/97285
Copyright and License Information
Copyright 2017 Eliana Duarte

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