FoCM 2014 conference
Workshop C1 - Computational Algebraic Geometry
December 18, 18:00 ~ 18:20 - Room B12
Generalized barycentric coordinates and algebraic geometry
Hal Schenck
University of Illinois, USA - schenck@math.uiuc.edu
Let $P_d$ be a convex polygon with $d$ vertices. The associated Wachspress surface $W_d$ is a fundamental object in approximation theory, defined as the image of the rational map $w_d$ from ${\mathbb P}^2$ to ${\mathbb P}^{d-1}$, determined by the Wachspress barycentric coordinates for $P_d$. We show $w_d$ is a regular map on a blowup $X_d$ of ${\mathbb P}^2$, and if $d>4$ is given by a very ample divisor on $X_d$, so has a smooth image $W_d$. We determine generators for the ideal of $W_d$, and prove that in graded lex order, the initial ideal of $I(W_d)$ is given by a Stanley-Reisner ideal. As a consequence, we show that the associated surface is arithmetically Cohen-Macaulay, of Castelnuovo-Mumford regularity two, and determine all the graded betti numbers of $I(W_d)$.
Joint work with Corey Irving (Santa Clara University, USA).