Syzygies and implicitization of tensor product surfaces

Duarte Gelvez, Eliana Maria

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

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

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

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

Copyright and License Information

Copyright 2017 Eliana Duarte

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