Combinatorics and quotients of toric varieties (Q1864185)

In the first part of the paper, the author studies linear projections of polytopes and dually projections of fans. A linear projection \(\pi\) of a full-dimensional polytope \(P\) in a real vector space \(V\) onto a polytope \(Q\) induces a polytopal subdivision of \(Q\), and the maximal cells of this subdivision are the chambers of \(Q\). Dually, one can consider the projection \(\pi^\vee\) from the dual vector space \(V^*\) to \(\text{ker}\,\pi^*\) obtained by dualizing the inclusion of the kernel of \(\pi\) into \(V\). The normal fan \(N(P)\) of the polytope \(P\) in \(V^*\) defines a fan \(\Delta\) in \(\text{ker}\,\pi^*\), namely the coarsest common refinement of all projections of cones in \(N(P)\) by \(\pi^\vee\). The fan \(\Delta\) is the normal fan of the fiber polytope of \(P\) and \(Q\) introduced by \textit{L. J. Billera} and \textit{B. Sturmfels} [Ann. Math. 135, 527-549 (1992; Zbl 0762.52003)]. The cones of \(\Delta\) are in one-to-one correspondence with the coherent strings of the projection \(\pi\), i.e. the coherent liftings of polytopal subdivisons of \(Q\) to \(P\). Yi Hu here extends this correspondence to a one-to-one correspondence between locally coherent strings of the projection \(\pi\) and what he calls virtual cones of the dual projection \(\pi^\vee\). Dually, he defines locally coherent costrings of the projection \(\pi^\vee\) as subfans of the normal fan of the polytope \(P\) that are liftings of fans in the image. Yi Hu proves that there is a one-to-one correspondence between locally coherent costrings of \(\pi^\vee\) and what he calls virtual chambers of the projection \(\pi\). The second part of this paper is devoted to the relation of the combinatorial constructions with toric geometry. For a given rational fan there is an associated toric variety, and a linear projection of the vector space containing the fan can be viewed as taking the quotient of the torus acting on the variety by a certain subgroup of the big torus. The author proves that an open toric subset of a toric variety admits a good quotient by a given subgroup if and only if the associated subfan is a locally coherent costring of the associated projection. This result was independently proved by \textit{H. A. Hamm} [CRC Res. Notes Math. 412, 61--75 (2000; Zbl 0949.14031)] and \textit{J. Świȩcicka} [Colloq. Math. 82, No.1, 105-116 (1999; Zbl 0961.14032)]. In the case of a projective toric variety corresponding to a polytopal fan, as a corollary of the correspondence between locally coherent costrings and virtual chambers, the author also obtains a characterization of open toric subvarieties with good quotients in terms of virtual chambers. This result generalizes the relation between G.I.T. quotients of projective toric varieties and chambers of projections of polytopes described by \textit{M. M. Kapranov}, \textit{B. Sturmfels} and \textit{A. V. Zelevinsky} [Math. Ann. 290, No. 4, 643--655 (1991; Zbl 0762.14023)].