Gale duality and homogeneous toric varieties

In this talk we shall discuss the results of the paper [1]. A non-degenerate toric variety $X$ is called $S$-homogeneous if the subgroup of $\mathrm{Aut}(X)$ generated by root subgroups acts on $X$ transitively. We shall introduce the notion of strongly regular fans, which correspond to $S$-homogeneous toric varieties. We shall discuss the correspondence between maximal $S$-homogeneous toric varieties and pairs $(P, A)$, where $P$ is an abelian group and $A$ is a finite admissible collection of elements of $P$, that generate $P$. From the explicit construction of a maximal $S$-homogeneous toric variety it can be deduced that for every non-degenerate homogeneous toric variety $X$ there is an open toric embedding of $X$ into a maximal $S$-homogeneous toric variety.

References:
[1] Ivan Arzhantsev. Gale duality and homogeneous toric varieties. Communications in Algebra, 46:8 (2018), 3539-3552