Projective geometry of Wachspress coordinates.

Kathlén Kohn (KTH, Stockholm)

19-Jun-2020, 12:30-13:30 (4 years ago)

Abstract: This talk brings many areas together: discrete geometry, statistics, intersection theory, classical algebraic geometry, and geometric modeling. First, we recall the definition of the adjoint of a polytope given by Warren in 1996 in the context of geometric modeling. He defined this polynomial to generalize barycentric coordinates from simplices to arbitrary polytopes. Secondly, we show how this polynomial appears in statistics. It is the numerator of a generating function over all moments of the uniform probability distribution on a polytope. Thirdly, we characterize the adjoint via a vanishing property: it is the unique polynomial of minimal degree which vanishes on the non-faces of a polytope. In addition, we see that the adjoint appears as the central piece in Segre classes of monomial schemes. Finally, we observe that adjoints of polytopes are special cases of the classical notion of adjoints of divisors. Since the adjoint of a simple polytope is unique, the corresponding divisors have unique canonical curves. In the case of three-dimensional polytopes, we show that these divisors are either K3- or elliptic surfaces. This talk is based on joint works with Kristian Ranestad, Boris Shapiro and Bernd Sturmfels

algebraic geometrydifferential geometrygeometric topologysymplectic geometry

Audience: researchers in the topic


The London geometry and topology seminar

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Organizer: Paolo Cascini*
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