Geometry of Wachspress surfaces

Let Pd 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 determined by the Wachspress barycentric coordinates for P_d. We show w_d is a regular map on a blowup X_d of ℙ² 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_Wd 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 IW_d