Cox Rings

Abstract

This is the first chapter of an introductory text under construction; further chapters are available on the author's web pages. Our aim is to provide an elementary access to Cox rings and their applications in algebraic and arithmetic geometry. We are grateful to Victor Batyrev and Alexei Skorobogatov for helpful remarks and discussions. Any comments and suggestions on this draft will be highly appreciated.

The Cox ring of a complexity-one horospherical variety

“…From the Luna-Vust theory [18], there is a combinatorial description of such varieties (see [23], [24, Ch. 16],and [17,§1]) which is quite similar to the classical one of torus embeddings (see, for instance, [8]). The geometry of complexity-one horospherical varieties has been studied in [17] …”

“…The combinatorics introduced thereafter are classifying objects for a specific category: the category of G-models of P 1 × G/H, whose objects are pairs (X, ψ), where X is a normal G-variety and ψ is a G-equivariant birational map as in (1). Morphisms (X 1 , ψ 1 ) → (X 2 , ψ 2 ) in this category are G-morphisms f : X 1 → X 2 such that ψ 2 • f = ψ 1 .…”

Euler sequence for complete smooth K⁎-surfaces

“…Since the classes of these curves generate the Mori cone of X (see [ADHL13]) we conclude that −K X is ample and thus X is del Pezzo. Moreover by [Hug13, Proposition 5.9] del Pezzo K * -surfaces without elliptic fixed points are toric.…”

A special Cayley octad

“…By definition, a class D on X is movable if the subset ∆∈|D| Supp(∆) has codimension at least 2 in X; the movable cone Mov(X) is the smallest closed cone in N 1 (X) containing all movable classes. For general properties of movable divisors and the movable cone, a reference is [ADHL,Section 3.3.2].…”