Syzygies and implicitization of tensor product surfaces
|Institution:||University of Illinois Urbana-Champaign|
|Keywords:||Implicitization; Syzygy; Rees algebras; Basepoints; Tensor product surface; Smooth quadric|
|Full text PDF:||http://hdl.handle.net/2142/97285|
A tensor product surface is the closure of the image of a rational map : P1 P1 >P3. These surfaces arise in geometric modeling and in this context it is useful to know the implicit equation of 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 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 are closely related to the geometry of the set of points at which is undefined.Advisors/Committee Members: Schenck, Henry (advisor), Nevins, Thomas (Committee Chair), Francis, George (committee member), Reznick, Bruce (committee member).