This note proves the existence of universal rational parametrizations. The description involves homogeneous coordinates on a toric variety coming from a lattice polytope. We first describe how smooth toric varieties lead to universal rational parametrizations of certain projective varieties. We give numerous examples and then discuss what happens in the singular case. We also describe rational ...

This is a tutorial on some aspects of toric varieties related to their potential use in geometric modeling. We discuss projective toric varieties and their ideals, as well as real toric varieties and the moment map. In particular, we explain the relation between linear precision and the moment map.

This is a tutorial on some aspects of toric varieties related to their potential use in geometric modeling. We discuss projective toric varieties and their ideals, as well as real toric varieties. In particular, we explain the relation between linear precision and a particular linear projection we call the algebraic moment map.

In this paper, we give a structure theorem for the derived category D(X) of a toric fibration X over P with fiber Y provided that Y has a full strongly exceptional collection of line bundles.

I work in combinatorics, algebraic geometry, convex geometry and commutative algebra while staying informed on certain topics in category theory and ring theory. In particular, I focus on toric varieties and singularity theory. The study of toric varieties lies at the intersection of combinatorics, algebraic geometry, convex geometry and integer programming. There is a correspondence between ce...

In today's increasingly heterogeneous compute landscape, there is high demand for design tools that offer seemingly contradictory features: portable programming abstractions hide underlying architectural detail, and the capability to optimise exploit features. Our meta-programming approach, Artisan, decouples application functionality from optimisation concerns address complexity of mapping hig...

The Oda’s Strong Factorization Conjecture states that a proper birational map between smooth toric varieties can be decomposed as a sequence of smooth toric blowups followed by a sequence of smooth toric blowdowns. This article describes an algorithm that conjecturally constructs such a decomposition. Several reductions and simplifications of the algorithm are presented and some special cases o...

In this paper we discuss combinatorial questions about lattice polytopes motivated by recent results on minimum distance estimation for toric codes. We also include a new inductive bound for the minimum distance of generalized toric codes. As an application, we give new formulas for the minimum distance of generalized toric codes for special lattice point configurations. 2010 MSC: 14M25, 14G50,...

Toric Ideals in Algebraic Statistics