Vol. 8, No. 2, 2014

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Geometry of Wachspress surfaces

Corey Irving and Hal Schenck

Vol. 8 (2014), No. 2, 369–396
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

${ℙ}^{2}\underset{}{\overset{{w}_{d}}{\to }}{ℙ}^{d-1},$

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 and of Castelnuovo–Mumford regularity $2$ and determine all the graded Betti numbers of ${I}_{{W}_{d}}$.

Keywords
barycentric coordinates, Wachspress variety, rational surface
Mathematical Subject Classification 2010
Primary: 13D02
Secondary: 52C35, 14J26, 14C20