Department of

January 2017 February 2017 March 2017 Su Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa 1 2 3 4 5 6 7 1 2 3 4 1 2 3 4 8 9 10 11 12 13 14 5 6 7 8 9 10 11 5 6 7 8 9 10 11 15 16 17 18 19 20 21 12 13 14 15 16 17 18 12 13 14 15 16 17 18 22 23 24 25 26 27 28 19 20 21 22 23 24 25 19 20 21 22 23 24 25 29 30 31 26 27 28 26 27 28 29 30 31

Friday, February 3, 2017

**Abstract:** A tensor product surface is the closure of the image of a map $\lambda:\mathbb{P}^1\times \mathbb{P}^1\to \mathbb{P}^3$. These surfaces arise in geometric modeling and in this context it is useful to know the implicit equation of $\lambda$ in $\mathbb{P}^{3}$. 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 $\lambda$ 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 $\lambda$ are closely related to the geometry of the set of points at which $\lambda$ is undefined.