# Projective geometry of Wachspress coordinates

## TL;DR

This paper explores the algebraic and geometric properties of Wachspress coordinates for convex polytopes, introducing a unique hypersurface related to the polytope's non-faces and connecting it to classical algebraic geometry.

## Contribution

It generalizes Wachspress's construction of adjoint curves to higher dimensions, linking it with the adjoint polynomial and algebraic geometry concepts.

## Key findings

- Identifies a unique hypersurface passing through non-faces of a polytope.
- Describes the Wachspress map and variety, including their algebraic properties.
- Classifies polytopes in 2D and 3D with special hypersurfaces, finding elliptic and K3 surfaces.

## Abstract

We show that there is a unique hypersurface of minimal degree passing through the non-faces of a polytope which is defined by a simple hyperplane arrangement. This generalizes the construction of the adjoint curve of a polygon by Wachspress in 1975. The defining polynomial of our adjoint hypersurface is the adjoint polynomial introduced by Warren in 1996. This is a key ingredient for the definition of Wachspress coordinates, which are barycentric coordinates on an arbitrary convex polytope. The adjoint polynomial also appears both in algebraic statistics, when studying the moments of uniform probability distributions on polytopes, and in intersection theory, when computing Segre classes of monomial schemes. We describe the Wachspress map, the rational map defined by the Wachspress coordinates, and the Wachspress variety, the image of this map. The inverse of the Wachspress map is the projection from the linear span of the image of the adjoint hypersurface. To relate adjoints of polytopes to classical adjoints of divisors in algebraic geometry, we study irreducible hypersurfaces that have the same degree and multiplicity along the non-faces of a polytope as its defining hyperplane arrangement. We list all finitely many combinatorial types of polytopes in dimensions two and three for which such irreducible hypersurfaces exist. In the case of polygons, the general such curves< are elliptic. In the three-dimensional case, the general such surfaces are either K3 or elliptic.