Some spaces associated with multigraded rings

Abstract

This thesis discusses multihomogeneous spaces and their relation with T-varieties and toric varieties. Firstly, we study multihomogeneous spaces corresponding to $\mathbb{Z}^n$-graded algebras over an algebraically closed field of characteristic $0$. A multihomogeneous space is a scheme associated with a graded ring where the graded group is an abelian group of finite rank. Geometrically, it is the geometric quotient of a quasi-open subscheme of the associated affine scheme by the corresponding diagonalisable group scheme. A scheme is divisorial if and only if it embeds into a multihomogeneous space. We give a criterion when a multihomogeneous space is normal. Then we mention that one could associate a sheaf with each graded module over the algebra, via a tilde construction, similar to the construction of a sheaf associated with a graded module over integer-graded rings. In doing so, we have a collection of shifted sheaves of modules associated with graded modules over algebra. As one can expect, this tilde construction is a covariant exact functor from the category of graded modules to the category of quasi-coherent sheaves of modules. We identify which shifted sheaves of modules are line bundles in terms of the graded group.

An affine T-variety is an affine scheme with an effective action of a torus. Such affine varieties can be represented by a proper polyhedral divisor over a semi projective variety. The semi projective variety is a good quotient of the action. A proper polyhedral divisor encodes a collection of ample Cartier divisors, some of which are big. We show that for an affine T-variety, the corresponding semi projective variety and the multihomogeneous space are birational. They are generally not isomorphic due to the lack of ample divisors on the multihomogeneous space. A toric variety is a T-variety such that the torus occurs as a dense open subscheme, and the action extends the multiplication of the torus. In toric varieties with enough invariant Cartier divisors, which includes simplicial toric varieties, points correspond to homogeneous prime ideals of a certain graded ring which Perling shows. His construction, known as tproj, reconstructs the toric variety from a graded ring where the graded group is the Picard group of the toric variety. We show that the construction of multihomogeneous space is similar to tproj; in fact, tproj, which is isomorphic to the toric variety, is an open subscheme of the multihomogeneous space associated with that graded ring. We give a criterion when a simplicial toric variety is a multihomogeneous space, and using this criterion, we classify all simplicial toric surfaces that are multihomogeneous spaces.