Some more toric Gorenstein varieties

As a follow-up to my previous post that encountered toric Gorenstein varieties on quite a general, abstract level I thought that I would write something much more computational. The result is my attempt to survey of some of the beautiful phenomenonology of Gorenstein toric varieties, starting with the classification of Gorenstein weighted projective spaces (equivalently, reflexive simplices). They turn out to biject with ‘unit partitions’

1=\frac{1}{q_0}+\dots+\frac{1}{q_n}

for q_i\in\mathbb{N}, with weighted projective spaces of dimension n corresponding to partitions of length n+1. These are related to the Sylvester sequence s_k defined by s_1=2 and

s_{k+1}=1+\prod_{m\leq k}s_m

There’s a famous result of Curtiss from 1921 stating that the largest possible value of q_i in a partition of length n is s_n-1 that, with a touch of toric geometry, yields bounds on the degree of weighted projective spaces (equivalently, volumes of reflexive simplices). Lastly, I describe a similarly famous result of Stanley stating that a reasonably general graded ring is Gorenstein iff the numerator of its Hilbert series is palindromic. As an immediate consequence, one sees that a Fano polytope is reflexive iff its Ehrhart series has a palindromic numerator. Wonderful.

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