Three-dimensional Algebraic Varieties Research Articles

The rationality problem for conic bundles

This expository paper is concerned with the rationality problem for three-dimensional algebraic varieties with a conic bundle structure. The main methods of this theory are discussed, proofs of certain principal results are sketched, and some recent achievements are presented. Many open problems are also stated. Bibliography: 209 titles.

On birational involutions of

We study elements of order two in the birational automorphism groups of rationally connected three-dimensional algebraic varieties such that there exists a non-uniruled divisorial component of the -fixed point locus. Using the equivariant minimal model program, we give a rough classification of such elements.

On the projectivity of threefolds

Let X be a smooth complete three-dimensional algebraic variety (defined over an algebraically closed field k). We show that X is projective if it contains a divisor which is positive on the cone of effective curves.

On Fibrations into del Pezzo Surfaces

The problem of birational rigidity for three-dimensional algebraic varieties fibered over rational curves into del Pezzo surfaces of degree 1 is discussed. A criterion for the rigidity of such fibrations in the Mori category is suggested and the inverse implication is proved (Theorem 3.3). Surgeries on fibers in fibrations of this type, which turn out to be closely related to the rigidity problem, are considered. In particular, an important result on the uniqueness of a smooth model in a class of maps over a base is stated (Corollary 4.5).

On the existence of complements of the canonical divisor for Mori conic bundles

This paper continues the author's study of extremal contractions in the sense of Mori from three-dimensional varieties onto surfaces. Such contractions occur in a natural way in the birational classification theory of three-dimensional algebraic varieties. Reid's “general elephant” conjecture of the complementedness of the canonical divisor and also the conjecture about singularities of the base surface are discussed. The situation is studied locally near a singular fibre.

ON THE CLOSED CONE OF CURVES OF ALGEBRAIC 3-FOLDS

In this paper the author establishes, under natural conditions, the local polyhedrality of the closed cone of curves of a three-dimensional algebraic variety in the part that is negative with respect to the canonical class. In particular, it is shown that there always exists an extremal ray giving a contraction. The results can be used in three-dimensional birational geometry. Bibliography: 10 titles.

Birational automorphisms of three-dimensional algebraic varieties

The survey is devoted to the birational theory of three-dimensional algebraic Fano varieties.

Anticanonical models of three-dimensional algebraic varieties

The survey reflects the present state of the biregular theory of three-dimensional algebraic varieties with negative canonical sheaf.

A FILTRATION IN THE THREE-DIMENSIONAL CREMONA GROUP

In this paper we introduce the concept of the genus of a birational mapping of nonsingular three-dimensional algebraic varieties over a field of characteristic 0 and prove that the automorphisms whose genus is at most a fixed number form a group. We also prove that a birational morphism of nonsingular three-dimensional varieties splits into a composition of a map which is an inverse to a morphism of genus 0 and monoidal transformations. Bibliography: 9 items.

Finite groups of birational selfmaps of threefolds

We determine three-dimensional algebraic varieties whose groups of birational selfmaps do not satisfy the Jordan property.