In Section 6.1, we give an overview of the theory of universal torsors over arbitrary fields of characteristic zero. Furthermore, we explore their connection to characteristic spaces and Cox rings. Section 6.2 deals with the existence of rational points over number fields. We discuss the Hasse principle and weak approximation. The failure of these principles is often explained by Brauer–Manin obstructions. We indicate how this can be approached via universal torsors and give an overview of the existing results. Section 6.3 is devoted to the distribution of rational points on varieties over number fields. Here, one introduces height functions on the set of rational points and is interested mainly in the asymptotic behavior of the number of rational points of bounded height. For Fano varieties, this behavior is predicted by Manin's conjecture. We indicate how universal torsors and Cox rings can be applied to prove Manin's conjucture. In Section 6.4, we specialize to del Pezzo surfaces. Here, Manin's conjecture is known in many cases, and a general strategy emerges. We discuss this strategy and some technical ingredients in detail. In Section 6.5, we show how it can be applied to prove Manin's conjecture for a singular cubic surface.

Universal torsors and Cox rings

Quasitori and principal homogeneous spaces

Here we summarize the basic concepts and facts on quasitori and their principal homogeneous spaces over nonclosed fields. The main references are [303, 304]. As an introduction to varieties over nonclosed fields and their rational points, we mention [253]. First we fix the setting.

We discuss various topics around Cox rings. The first section is devoted to birational maps, for example, blow-ups. We figure out the class of modifications that preserve finite generation and we show how to compute the Cox ring of the modified variety in terms of the Cox ring of the initial one. In Section 4.2, we first introduce a class of quotient presentations dominated by the characteristic space and comprising, for example, the classical affine cones. Then we provide a lifting result for connected subgroups of the automorphism group. In the case of a finitely generated Cox ring, this gives an approach to the whole automorphism group, showing in particular that it is affine algebraic. The topic of Section 4.3 is finite generation of the Cox ring. We provide general criteria and characterizations; for example, the finiteness characterization of Hu and Keel in terms of polyhedral subdivisions of the moving cone is proven and we discuss finite generation of the Cox ring for Fano varieties. In Section 4.4 we consider varieties coming with a torus action. We express their Cox ring in terms of data of the action. In the case of a rational variety with an action of complexity 1, we see that the Cox ring is given by trinomial relations as in Section 3.4. In particular, the constructions given there produce indeed all rational normal complete A2-varieties with a torus action of complexity 1. Section 4.5 is about almost homogeneous varieties, that is, equivariant open embeddings of homogeneous spaces. We first describe the Cox ring of a homogeneous space G/H. Embeddings X of homogeneous spaces G/H with finitely generated Cox ring and small boundary X \ G/H allow an immediate description via bunched rings and provide a rich example class. Finally, we survey in this section results on the case of wonderful and more general spherical varieties.

Toric varieties form an important class of examples in algebraic geometry, as they admit a complete description in terms of combinatorial data, so-called lattice fans. In Section 2.1, we briefly recall this description and also some of the basic facts in toric geometry. Then we present Cox's construction of the characteristic space of a toric variety in terms of a defining fan and discuss the basic geometry around this. Section 2.2 is pure combinatorics. We introduce the notion of a “bunch of cones” and show that, in an appropriate setting, this is the Gale dual version of a fan. Under this duality, the normal fans of polytopes correspond to bunches of cones arising canonically from the chambers of the so-called Gelfand–Kapranov–Zelevinsky decomposition. In Section 2.3, we discuss the geometric meaning of bunches of cones: they encode the maximal separated good quotients for subgroups of the acting torus on an affine toric variety. In Section 2.4, we specialize these considerations to toric characteristic spaces, that is, to the good quotients arising from Cox's construction. This leads to an alternative combinatorial description of toric varieties in terms of “lattice bunches,” which turns out to be particularly suitable for phenomena around divisors.

Toric varieties

Toric varieties and fans

We introduce toric varieties and their morphisms and recall that this category admits a complete description in terms of lattice fans.

Definition 2.1.1.1 A toric variety is an irreducible, normal variety X together with an algebraic torus action T × X → X and a base point x0∈ X such that the orbit map T → X, t → t ·x0 is an open embedding.