Toric quotients and flips

As my contribution to these proceedings, I will discuss the geometric invariant theory quotients of toric varieties. Specifically, I will show that quotients of the same problem with respect to different linearizations are typically related by a sequence of flips (or more precisely, log flips) in the sense of Mori, which in the quasi-smooth case can be characterized quite precisely. This seems to have little to do with the subject of the symposium, which was low-dimensional topology. However, it is intended as a model of a general theory, to be described elsewhere [11, 14], on the dependence of invariant theory quotients on linearizations. This theory was the subject of my talk in the symposium, and it can be applied to many of the moduli spaces employed in low-dimensional topology. Since we will be studying geometric invariant theory, we will want all of our toric varieties to be quasi-projective. Accordingly, we will define them using polyhedra, which is essentially dual to the usual approach involving fans. This dual approach is much better suited to our own purposes, but it is not well documented in the literature. The whole of §2 is therefore expository, presenting well-known results in a polyhedral context. The usual approach, and its relationship with ours, are explained in the excellent new book of Fulton [3]. The bulk of the paper concerns the quotients of an arbitrary quasi-projective toric variety by a subtorus of the usual torus action. We show in §3 that the linearizations giving nonempty quotients are parametrized by a polyhedron (in fact, the projection on a subspace of the polyhedron defining the toric variety), and that the polyhedron is partitioned into polyhedral chambers (bounded by the projection of the appropriate skeleton of the polyhedron) inside which the quotient is essentially constant. Furthermore, we show that moving between adjacent chambers induces a birational map of the quotients which, in good cases, is a flip. Though they are important for us, these results are relatively easy. Indeed, though we prove them by toric methods, they also essentially follow from the descent lemma and the numerical criterion. The main result, (4.5), is stronger, and correspondingly harder, occupying most of §4. It gives an explicit description of the flip as a weighted blow-up and blow-down in the case when the toric variety is quasi-smooth, that is, a finite abelian quotient of a smooth variety. In the bad cases where the birational map is not a flip, either the blow-down or the blow-up is absent. These results are proved by a series of simplifications leading to an easy model case. Three existing papers are very closely related to the present one. First, Kapranov, Sturmfels and Zelevinsky [7] have studied quotients of toric varieties by subtori, though their interest is more in the Chow quotients, and their relation to the inverse system of geometric invariant theory quotients, than in the matters we treat. Second, Guillemin and Sternberg [4] have carried out the whole program of §4 in the symplectic category, indeed treating arbitrary symplectic manifolds with torus actions, not just toric varieties. Finally, a recent paper of Hu [6] proves part of our (4.5)(a), in which we identify the exceptional loci of the birational maps between quotients, for torus actions on arbitrary