#### 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
##### Milestones
Received: 6 January 2013
Revised: 7 April 2013
Accepted: 27 May 2013
Published: 18 May 2014
##### Authors
 Corey Irving Department of Mathematics and Computer Science Santa Clara University 500 El Camino Real Santa Clara, CA 95053 United States Hal Schenck Department of Mathematics University of Illinois at Urbana–Champaign 1409 West Green Street Urbana, IL 61801 United States