A footnote to a theorem of Kawamata

Kawamata has shown that the quasi-Albanese map of a quasi-projective variety with log-irregularity equal to the dimension and log-Kodaira dimension 0 is birational. In this note, we show that under these hypotheses the quasi-Albanese map is proper in codimension 1 as conjectured by Iitaka.     

A perturbation result for q-open quotient morphisms in normed spaces and applications to linear relations

Using some techniques of perturbation theory for quotient morphisms between Banach spaces we obtain necessary and sufficient conditions for the stability of the topological index of a q-open quotient morphism under small (with respect to the gap topology) perturbations with quotient morphisms. As an application we obtain necessary and sufficient conditions for the stability of the topological index of an open linear relation with closed multivalued part between normed spaces under small perturbations with linear relations.     

Induced nilpotent orbits and birational geometry

In general, a nilpotent orbit closure in a complex simple Lie algebra g, does not have a crepant resolution. But, it always has a Q-factorial terminalization by the minimal model program. According to B. Fu, a nilpotent orbit closure has a crepant resolution only when it is a Richardson orbit, and the resolution is obtained as a Springer map for it. In this paper, we shall generalize this result to Q-factorial terminalizations when g is classical. Here, the induced orbits play an important role instead of Richardson orbits.     

拟正则双序集用矩形双序集的余扩张

称双序集E为双序集F用矩形双序集的余扩张,若存在满双序集态射θ：E→F,使对每个α∈F,αθ－1是E的矩形双序子集．本文讨论了拟正则双序集的这种余扩张的性质,给出了它们的结构．作为应用,证明了拟正则的硬双序集实为正则双序集．     

Left almost split morphisms and generalized weyl algebras

It is proved that the category of modules of finite length over a broad class of generalized Weyl algebras contains no left almost split morphism starting from a simple module. It is shown that a similar assertion holds for the algebraUsl₂(k) over an algebraically closed field k of characteristic 0. As a by-product, a new series of simple modules for such algebras is constructed. Translated fromMatematicheskie Zametki, Vol. 66, No. 5 pp. 734–740, November, 1999.     

Triple Brackets and Lax Morphism Categories

Various aspects of the traditional homotopy theory of topological spaces may be developed in an arbitrary 2-category C with zeros. In particular certain secondary composition operations called box brackets recently have been defined for C; these are similar to, but extend, the familiar Toda brackets in the topological case. In this paper we introduce further the notion of a suspension functor in C and explore the ramifications of relativizing the theory in terms of the associated lax morphism category of C, denoted mC. Four operations associated to a 3-box diagram are introduced and relations among them are clarified. The results and insights obtained, while by nature somewhat technical, yield effective and efficient techniques for computing many operations of Toda bracket type. We illustrate by recording some computations from the homotopy groups of spheres. Also the properties of a new operation, the 2-sided matrix Toda bracket, are explored.     

Auslander–Reiten Components Containing Cones

Let A be a locally finite Abelian R-category with Auslander–Reiten sequences and with Auslander–Reiten quiver (A). We give a criterion for Auslander–Reiten components to contain a cone and apply this result to various categories.     

Connectedness with Respect to a Closure Operator

A notion of connectedness with respect to a closure operator C and a class of monomorphisms N is introduced in an arbitrary category X. It is shown that under appropriate hypotheses, most classical results about topological connectedness can be generalized to this setting. Examples that illustrate this new concept are provided.     

On resolving singularities of plane curves via a theorem attributed to Alfred Clebsch

This paper discusses a theorem in birational geometry that J.L. Coolidge attributed to Alfred Clebsch. The background is reconstructed from letters Felix Klein exchanged with Max Noether in 1894, when Noether was completing work on a lengthy report with Alexander Brill on the history of algebraic functions. Noether was deeply troubled to learn that Klein had informed him back in 1869 about relevant results that Clebsch and Leopold Kronecker had discussed in Berlin. These exchanges with Klein led to revisions in the Brill-Noether report, made in part to ensure Noether's own priority rights and larger intellectual legacy.