Geometry of Wachspress surfaces

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 mapdetermined by the Wachspress barycentric coordinates for P d .We show w d is a regular map on a blowup X d of ‫ސ‬ 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 .

Locations

• Algebra & Number Theory - View - PDF
• arXiv (Cornell University) - View - PDF
• Project Euclid (Cornell University) - View - PDF