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