Geometry of Wachspress surfaces

TL;DR

This paper explores the geometric and algebraic properties of Wachspress surfaces derived from convex polygons, establishing their regularity, ideal generators, and Cohen-Macaulay status, with detailed Betti number computations.

Contribution

It provides a detailed algebraic description of Wachspress surfaces, including their ideal generators, regularity, and Betti numbers, and shows they are arithmetically Cohen-Macaulay.

Findings

Wachspress map extends to a smooth surface for d>4

Initial ideal of the Wachspress surface's ideal is a Stanley-Reisner ideal

Wachspress surfaces are arithmetically Cohen-Macaulay with regularity two

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 w_d from P^2 to 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 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).