Geometry of Wachspress surfaces

Abstract

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

$${\mathbb{P}}^2\underset{}{\overset{w_d}{\to }}{\mathbb{P}}^{d-1},$$

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 and of Castelnuovo–Mumford regularity $2$ and determine all the graded Betti numbers of $I_{W_d}$.