Toric Gorenstein varieties

It’s been sometime since I said that I was going to start a series on the ‘foundations’ of toric geometry. In many ways toric geometry is the antithesis of a ‘foundational’ subject: its primary benefits are ease of computation and experimentation, and the point is that it shifts technical difficulty into combinatorics and away from less accessible areas. Anyway, I wrote some notes adapted from the book of Cox-Little-Schenk.

The main focus of the notes is on toric Fano varieties and toric Gorenstein varieties. I’ve already expounded on the former quite regularly, so I will make some brief comments about the latter. From a high-level standpoint, Gorenstein varieties are the possibly singular varieties one would consider in order to try to build up a reasonable duality theory similar to the smooth case. They are the varieties for which the dualising sheaf \omega_X is a line bundle, and hence for which K_X is Cartier. There’s of course a purely algebraic incarnation of this as ‘Gorenstein rings’, which were studied by Grothendieck, Serre, famously by Bass, and by many others in the mid-20th century.

Stanley and others pioneered the extraordinarily ‘symmetric’ combinatorics of Gorenstein rings and varieties from the 1970s onwards; I make some remarks in my notes. One particular claim that I am quite fond of is the following. Toric Gorenstein Fano varieties correspond to ‘reflexive polytopes’. There are 16 reflexive polygons – it’s quite fun to find them all! – and they all have the property that the sum of the number of their boundary lattice points and the number of boundary lattice points on their dual is 12. There’s a geometrical proof of this via Noether’s formula (or RR for smooth surfaces):

\chi(\mathcal{O}_X)=\frac{K_X^2+e(X)}{12}

where e(X) is the topological Euler characteristic. Since \chi(\mathcal{O}_X)=1 for a toric variety, there’s the 12. The other two terms turn out to be given by the two boundary lattice point counts I mentioned.

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