Jürgen Hausen

Toric Prevarieties and Subtorus Actions

AbstractDropping separatedness in the definition of a toric variety, one obtains the more general notion of a toric prevariety. Toric prevarieties occur as ambient spaces in algebraic geometry and, moreover, they appear naturally as intermediate steps in quotient constructions. We first provide a complete description of the category of toric prevarieties in terms of combinatorial data, so-called systems of fans. In a second part, we consider actions of subtori H of the big torus of a toric prevariety X and investigate quotients for such actions. Using our language of systems of fans, we characterize existence of good prequotients for the action of H on X. Moreover, we show by means of an algorithmic construction that there always exists a toric prequotient for the action of H on X, that means an H-invariant toric morphism p from X to a toric prevariety Y such that every H-invariant toric morphism from X to a toric prevariety factors through p. Finally, generalizing a result of D. Cox, we prove that every toric prevariety occurs as the image of a categorical prequotient of an open toric subvariety of some

On embeddings into toric prevarieties

\mathbb{C}

Gluing Affine Torus Actions Via Divisorial Fans

s.

Examples and Counterexamples for Existence of Categorical Quotients

Generalizing cones over projective toric varieties, we present arbitrary toric varieties as quotients of quasiaffine toric varieties. Such quotient presentations correspond to groups of Weil divisors generating the topology. Groups comprising Cartier divisors define free quotients, whereas ℚ–Cartier divisors define geometric quotients. Each quotient presentation yields homogeneous coordinates. Using homogeneous coordinates, we express quasicoherent sheaves in terms of multigraded modules and describe the set of morphisms into a toric variety.

On Terminal Fano 3-Folds with 2-Torus Action

We give examples of complete normal surfaces that are not em- beddable into simplicial toric prevarieties nor toric prevarieties of ane inter- section.

Factorial algebraic group actions and categorical quotients

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 Coxs 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 Coxs 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 x 0 ∈ X such that the orbit map T → X , t → t · x 0 is an open embedding.