Cox ring

Abstract: This paper will introduce the homogeneous coordinate ring S of a toric variety X. The ring S is a polynomial ring with one variable for each one-dimensional cone in the fan ∆ determining X, and S has a natural grading determined by the monoid of effective divisor classes in the Chow group A n−1 (X) of X (where n = dim X). Using this graded ring, we will show that X behaves like projective space in many ways. The paper is organized into four sections as follows. In §1, we define the homogeneous coordinate ring S of X and compute its graded pieces in terms of global sections of certain coherent sheaves on X. We also define a monomial ideal B ⊂ S that describes the combinatorial structure of the fan ∆. In the case of projective space, the ring S is just the usual homogeneous coordinate ring C[x 0 ,. .. , x n ], and the ideal B is the \" irrelevant \" ideal x 0 ,. .. , x n. Projective space P n can be constructed as the quotient (C n+1 −{0})/C *. In §2, we will see that there is a similar construction for any toric variety X. In this case, the algebraic group G = Hom Z (A n−1 (X), C *) acts on an affine space C ∆(1) such that the categorical quotient (C ∆(1) − Z)/G exists and is isomorphic to X. The exceptional set Z is the zero set of the ideal B defined in §1. If X is simplicial (meaning that the fan ∆ is simplicial), then X ≃ (C ∆(1) − Z)/G is a geometric quotient, so that elements of C ∆(1) − Z can be regarded as \" homogeneous coordinates \" for points of X. For any toric variety, we will see in §3 that finitely generated graded S modules give rise to a coherent sheaves on X, and when X is simplicial, every coherent sheaf arises in this way. In particular, every closed subscheme of X is determined by a graded ideal of S. We will also study the extent to which this correspondence fails to be is one-to-one. Another feature of P n is that the action of P GL(n + 1, C) on P n lifts to an action of GL(n + 1, C) on C n+1 − {0}. In §4, we will see that …

Abstract: This is the first chapter of an introductory text under construction; further chapters are available via the authors' web pages Our aim is to provide an elementary access to Cox rings and their applications in algebraic and arithmetic geometry Any comments and suggestions on this draft will be highly appreciated

Abstract: Our main result is the description of generators of the total coordinate ring of the blow-up of P n in any number of points that lie on a rational normal curve. As a corollary we show that the algebra of invariants of the action of a two-dimensional vector group introduced by Nagata is finitely generated by certain explicit determinants. We also prove the finite generation of the algebras of invariants of actions of vector groups related to T-shaped Dynkin diagrams introduced by Mukai.