Quotients of toric varieties by the action of a subtorus

We consider the action of a subtorus of the big torus on a toric variety.The aim of the paper is to define a natural notion of a quotient for this setting and to give an explicit algorithm for the construction of this quotient from the combinatorial data corresponding to the pair consisting of the subtorus and the toric variety.Moreover, we study the relations of such quotients with good quotients.We construct a good model, i.e. a dominant toric morphism from the given toric variety to some "maximal" toric variety having a good quotient by the induced action of the given subtorus. Introduction.Let X be an algebraic variety with a regular action of an algebraic group G.A categorical quotient is a morphism p: X-^ Y which is G-invariant (i.e.constant on G-orbits) and satisfies the following universal property: every G-invariant morphism /: X-+Z factors uniquely through/?(see [Mu; Fo; Ki]).Though this universal property seems to be a minimal requirement for a quotient, there is no hope for the general existence of categorical quotients.(See e.g.[AC; Ha] for an explicit example of a C*-action on a smooth four-dimensional toric variety which does not have a categorical quotient, even if one allows the quotient space Y to be an algebraic or analytic space.)In the present article we consider toric varieties X with an action of an algebraic torus //; we refer to these varieties as toric //-varieties.The specialization of the definition of the categorical quotient to the category of toric varieties leads to the following notion: We call a toric morphism p : X-> Y a toric quotient, if it is //-invariant and every //-invariant toric morphism factors uniquely through p.For this kind of quotient we can actually prove the existence (see Theorem 1.4):For every toric H-variety X there exists a toric quotient.Our proof of this result is constructive.In fact, we introduce the notion of a quotient fan of a fan by some sublattice (see Section 2) and give an algorithm for the calculation of this quotient fan.We obtain the existence of toric quotients by applying this algorithm to the fan Δ of X and the lattice L of one-parameter subgroups of the acting torus T of X factoring through H.A particularly important notion of quotient is the so-called good quotient (see [Se]) generalizing the quotients occuring in Mumford's geometric invariant theory for project!ve varieties.Unfortunately, good quotients exist only under very special 1991