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

Let Pd be a convex polygon with d vertices. The associated Wachspress surface Wd is a fundamental object in approximation theory, defined as the image of the rational map

2 w dd1,

determined by the Wachspress barycentric coordinates for Pd. We show wd is a regular map on a blowup Xd of 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 and of Castelnuovo–Mumford regularity 2 and determine all the graded Betti numbers of IWd.

barycentric coordinates, Wachspress variety, rational surface
Mathematical Subject Classification 2010
Primary: 13D02
Secondary: 52C35, 14J26, 14C20
Received: 6 January 2013
Revised: 7 April 2013
Accepted: 27 May 2013
Published: 18 May 2014
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