GEOMETRY OF WACHSPRESS SURFACES

Let Pd be a convex polygon with d vertices. The associated Wach- spress surface Wd is a fundamental object in approximation theory, defined as the image of the rational map P 2 wd ! P d 1 , determined by the Wachspress barycentric coordinates for Pd. We show wd is a regular map on a blowup Xd of P 2 and if d > 4 is given by a very ample divisor on Xd, so has a smooth image Wd. We determine generators for the ideal of Wd, and prove that in graded lex order, the initial ideal of IWd 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 IWd .