Syzygies and implicitization of tensor product surfaces

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.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2017

Author dc:creator

• Duarte Gelvez, Eliana Maria

Contributors dc:contributor

• Schenck, Henry
• Nevins, Thomas
• Francis, George
• Reznick, Bruce

• Implicitization
• Syzygy
• Rees algebras
• Basepoints
• Tensor product surface
• Smooth quadric

Statement dc:rights

• Copyright 2017 Eliana Duarte

Language dc:language

en

Handle dc:identifier

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

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/97285