O ct 2 00 5 All the GIT quotients at once

Let T be an algebraic torus acting on a smooth variety V . We study the relationship between the various GIT quotients of V and the symplectic quotient of the cotangent bundle of V . 1 The general case Let V be a smooth algebraic variety over an arbitrary field k. We will assume that V is projective over affine, which means that the natural map V → SpecOV is projective. Let T be an algebraic torus over k acting effectively on V , and let L be a T -equivariant ample line bundle on V . A point p ∈ V is called L-semistable if there exists a T -invariant section of a positive power of L that does not vanish at p. The set of L-semistable points of V will be denoted V (L). An L-semistable point p is called L-stable if T acts locally freely at p and its orbit is closed in V (L). The set of L-stable points of V will be denoted V (L). If every L-semistable point is L-stable, then we will call L nice. We will consider two equivariant ample line bundles to be equivalent if they induce the same stable and semistable sets. Let {Li | i ∈ I} be a complete set of representatives of equivalence classes of nice line bundles with nonempty stable sets. Let V lf denote the set of points of V at which T acts locally freely. By definition, V (L) is contained in V lf for any L. The following lemma is a converse to this fact. Lemma 1.1 Suppose that there exists at least one ample equivariant line bundle on V . If T acts locally freely at p, then p is L-stable for some nice L, thus we have V lf = ⋃