The Cone of Semisimple Monoids with the same Factorial Hull

The factorial hull of the projective variety X (or its cone) is a graded algebra R(X) that can be used in some situations to consider simultaneously all divisor classes on X. In this paper we consider initially the situation where X is a semisimple variety associated with the semisimple monoid M . The factorial hull of such X is determined by a certain arrangement H of hyperplanes in the space of rational characters X(T )⊗Q of a maximal torus T of G0. If G0 is simply connected R(X) is the coordinate ring of Vinberg’s enveloping monoid Env(G0). Associated withX is a certain cone H ⊆ Cl(X) in the class group of X. Each δ ∈ H corrresponds to a semisimple monoid Mδ with R(Mδ) = R(X). M and N have the same factorial hull if XM and XN differ by G×Gorbits of codimension two or more. We calculate H explicitly in the case where X is the plongement magnifique for the simple goup G0. This is exactly the case where M is a canonical monoid. 1 The Q-Factorial Hull The Q-factorial hull of a projective variety is easy to describe, assuming it exists. This construction is related to a well-known question of Hilbert [11, 13, 18, 31]. Assume that X is an irreducible, normal, projective variety over the algebraically closed fieldK of characteristic zero. Let Cl(X) be the divisor class group ofX, and assume that F ⊆ Cl(X) is a free abelian subgroup of finite rank. Choose representatives Mα, α ∈ F , consisting of rank-one, locally reflexive sheaves on X. By results of [4, 1, 5], there is a natural graded K-algebra structure on RF (X) = ⊕