UC Berkeley

Commutative Algebra and Algebraic Geometry Seminar

November 27, 2018

939 Evans Hall

3:45PM: Ehrhart positivity and McMullen's formula

Fu Liu

The Ehrhart polynomial counts the number of integral points inside dilations of an integral polytope, that is, a polytope whose vertices are integral points. We say a polytope is Ehrhart positive if its Ehrhart polynomial has positive coefficients. In the literature, different families of polytopes have been shown to be Ehrhart positive using different techniques. We will survey these results in the first part of the talk, after giving a brief introduction to polytopes and Ehrhart polynomials. Through work of Danilov/McMullen, there is an interpretation of Ehrhart coefficients relating to the normalized volumes of faces. In the second part of the talk, I will discuss joint work with Castillo in which we try to make this relation more explicit in the case of regular permutohedra. The motivation is to prove Ehrhart positivity for generalized permutohedra. This turns out to be related to formulas for Todd classes of a certain family of toric varieties.

5:00PM: Canonical strip hypotheses for orbifolds

Ben Wormleighton

The Hilbert polynomial of a polarised variety is a well-studied invariant, but one that has many more secrets to give up. In 2009 Golyshev shared some insight into the structure of the roots of the Hilbert polynomial of a smooth Fano variety, inspired by work of Rodriguez-Villegas on generating functions, Yau on constraints for characteristic classes, and by several others on roots of Ehrhart polynomials. The shape of the results proven and conjectured in these papers became known as ‘canonical strip hypotheses’. I will outline these hypotheses, share recent counterexamples due to Belmans, Galkin, and Mukhopadhyay, and describe some of my recent work in extending these hypotheses to the orbifold setting.

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