— In this paper we show how to recover a special class of valuations (which generalize in a natural way the Zariski prime divisors) of function fields from the Galois theory of the functions fields in discussion. These valuations play a central role in the birational anabelian geometry and related questions. Résumé (Théorie de Galois pro-` des diviseurs premiers de Zariski) Dans cet article nou...

It is shown that the ideal theories of the fields of all meromorphic functions on any two noncompact Riemann surfaces are isomorphic. Further, various new representation and factorization theorems are proved. Introduction. Throughout this paper let X and Y denote noncompact (connected) Riemann surfaces. Let A(X) (or A for short), denote the ring of all analytic functions on X, and let F(X) (or ...

We give a cohomological characterization of semiample line bundles. The result is a common generalization of the Fujita-Zariski Theorem on semiampleness and the Grothendieck-Serre criterion for ampleness. As an application of the Fujita-Zariski Theorem we characterize contractible curves in 1-dimensional families.

Let G be a semi-simple Lie group and Γ < G be a lattice. This paper is motivated by the attempt to understand the infinite index subgroup structure of Γ . In particular, to understand the possibilities for infinite index, finitely generated, freely indecomposable, Zariski dense subgroups of Γ . The study of Zariski dense subgroups of semi-simple Lie groups has a long and rich history. Some high...

In the classical case all unirational surfaces are rational. This was rst realized by Oscar Zariski ((20], p. 314). Prompted by Hironaka's suggestion, we began an investigation of the type of surfaces introduced by Zariski in that paper. The research was originally begun by Blass in 1970-71 at Harvard with the advice of Hironaka and Zariski, and then during 1974{1977 he continued under the dire...

Title: The Dixmier-Moeglin equivalence for D-groups Abstract: The Dixmier-Moeglin equivalence is a characterization of the primitive ideals of an algebra that holds for many classes of rings, including affine PI rings, enveloping algebras of finite-dimensional Lie algebras, and many quantum algebras. For rings satisfying this equivalence, it says that the primitive ideals are precisely those pr...

In this paper, the authors study when the closure (in the Zariski topology) of orbits of representations of quivers of type A are rationally smooth. This is done by considering the corresponding quantized enveloping algebra U and studying the action of the bar involution on PBW bases. Using Ringel’s Hall algebra approach to quantized enveloping algebras and also AuslanderReiten quivers, we can ...

We formulate and prove a generalization of Zariski-van Kampen theorem on the topological fundamental groups of smooth complex algebraic varieties. As an application, we prove a hyperplane section theorem of Lefschetz-Zariski-van Kampen type for the fundamental groups of the complements to the Grassmannian dual varieties.

4 Zariski sheaves associated with pretheories. 25 4.1 Technical lemmas. . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.2 Pretheories in a neighborhood of a smooth subscheme. . . . . . 31 4.3 Pretheories on curves over a field. . . . . . . . . . . . . . . . . 35 4.4 Pretheories on semi-local schemes. . . . . . . . . . . . . . . . . 36 4.5 Zariski cohomology of sheaves associated with pre...