Zariski decomposition of b-divisors

Based on a recent work of Thomas Bauer’s (J. Algebr. Geom., to appear), reproving the existence of Zariski decompositions for surfaces, we construct a b-divisorial analogue of Zariski decomposition in all dimensions.     

Completeness in Zariski groups

Zariski groups are ℵ₀-stable groups with an axiomatically given Zariski topology and thus abstract generalizations of algebraic groups. A large part of algebraic geometry can be developed for Zariski groups. As a main result, any simple smooth Zariski group interprets an algebraically closed field, hence is almost an algebraic group over an algebraically closed field.     

Discrminantal groups and Zariski pairs of sextic curves

A series of Zariski pairs and three Zariski triplets were found by using lattice theory of K3 surfaces. Explicit equations for typical ones were calculated.     

Convexity and Zariski decomposition structure

This is the first part of our work on Zariski decomposition structures, where we study Zariski decompositions using Legendre–Fenchel type transforms. In this way we define a Zariski decomposition for curve classes. This decomposition enables us to develop the theory of the volume function for curves defined by the second named author, yielding some fundamental positivity results for curve classes. For varieties with special structures, the Zariski decomposition for curve classes admits an interesting geometric interpretation.     

Zariski surfaces, class groups and linearized systems

Galois descent and linearized systems in characteristic p≠0 are applied to the study of divisor class groups of Zariski surfaces. An efficient method is obtained for calculating their class groups and, for a general case, an easily calculated matrix is defined whose rank determines the class group order.     

Cubic surfaces violating the Hasse principle are Zariski dense in the moduli scheme

We construct new examples of cubic surfaces, for which the Hasse principle fails. Thereby we show that, over every number field, the counterexamples to the Hasse principle are Zariski dense in the moduli scheme of non-singular cubic surfaces.     

Zariski geometries

A characterization is obtained of the Zariski topology over an algebraically closed field.

     

An Analytic Zariski Structure Over a Field

Following the introduction and preliminary investigations of analytic Zariski structures in Peatfield and Zilber (Ann pure Appl Logic 132:125–180, 2005) an example of an analytic Zariski structure extending an algebraically closed field is provided. The example is constructed using Hrushovski’s method of free amalgamation, and a topology is introduced in which we can verify the analytic Zariski axioms.     

A Zariski Topology for Bicomodules and Corings

In this paper we introduce and investigate top (bi)comodules of corings, that can be considered as dual to top (bi)modules of rings. The fully coprime spectra of such (bi)comodules attains a Zariski topology, defined in a way dual to that of defining the Zariski topology on the prime spectra of (commutative) rings. We restrict our attention in this paper to duo (bi)comodules (satisfying suitable conditions) and study the interplay between the coalgebraic properties of such (bi)comodules and the introduced Zariski topology. In particular, we apply our results to introduce a Zariski topology on the fully coprime spectrum of a given non-zero coring considered canonically as duo object in its category of bicomodules. Supported by King Fahd University of Petroleum & Minerals, Research Project # INT/296.     

Zariski Pairs of Index 19 and Mordell-Weil Groups of K3 Surfaces

We find Zariski pairs of sextics with simple singularities havingmaximal total Milnornumber. We also relate them to the existenceof distinct Mordell–Weil groups of extremal elliptic K3surfaces with a fixed set of semistable singular fibres. 1991Mathematics Subject Classification: 14F45, 14F27, 14F28.     

A dual Zariski topology for modules

We introduce a dual Zariski topology on the spectrum of fully coprime R-submodules of a given duo module M over an associative (not necessarily commutative) ring R. This topology is defined in a way dual to that of defining the Zariski topology on the prime spectrum of R. We investigate this topology and clarify the interplay between the properties of this space and the algebraic properties of the module under consideration.     

On existence of log minimal models and weak Zariski decompositions

We first introduce a weak type of Zariski decomposition in higher dimensions: an -Cartier divisor has a weak Zariski decomposition if birationally and in a numerical sense it can be written as the sum of a nef and an effective -Cartier divisor. We then prove that there is a very basic relation between Zariski decompositions and log minimal models which has long been expected: we prove that assuming the log minimal model program in dimension d − 1, a lc pair (X/Z, B) of dimension d has a log minimal model (in our sense) if and only if K _X  + B has a weak Zariski decomposition/Z.     

Zariski closure, completeness and compactness

The categorical theory of closure operators is used to introduce and study separated, complete and compact objects with respect to the Zariski closure operator naturally defined in any category X(A,Ω) obtained by a given complete category X (endowed with a proper factorization structure for morphisms) and by a given X-algebra (A,Ω) by forming the affine X-objects modelled by (A,Ω). Several basic examples are provided.     

Zariski extensions and biregular rings

In this paper we study the structure of Zariski central rings with regular center i.p. biregular rings, and we obtain structure theorems for algebras which are finitely generated over their regular center, etc. Characterizations of certain classes of rings are being obtained by using localization at prime ideals and local-global theorems.     

Strong approximation for Zariski dense subgroups over arbitrary global fields

Consider a finitely generated Zariski dense subgroup of a connected simple algebraic group G over a global field F. An important aspect of strong approximation is the question of whether the closure of in the group of points of G with coefficients in a ring of partial adeles is open. We prove an essentially optimal result in this direction, based on the condition that is not discrete in that ambient group. There are no restrictions on the characteristic of F or the type of G, and simultaneous approximation in finitely many algebraic groups is also studied. Classification of finite simple groups is not used. Received: August 31, 1999.     

Gorenstein Modules over Zariski Filtered Rings

Abstract

We study Gorenstein injective and projective modules over Zariski filtered rings and obtain relations between the Gorenstein dimensions on the category of filtered modules from the associated category of graded modules over the associated graded ring.     

A survey on Zariski cancellation problem

In this survey article we describe known results and open questions on the Zariski cancellation problem, highlighting recent developments on the problem. We also discuss its close relationship with some of the other central problems on polynomial rings.     

Divisorial Zariski decompositions on compact complex manifolds

Using currents with minimal singularities, we introduce pointwise minimal multiplicities for a real pseudo-effective (1,1)-cohomology class α on a compact complex manifold X, which are the local obstructions to the numerical effectivity of α. The negative part of α is then defined as the real effective divisor N(α) whose multiplicity along a prime divisor D is just the generic multiplicity of α along D, and we get in that way a divisorial Zariski decomposition of α into the sum of a class Z(α) which is nef in codimension 1 and the class of its negative part N(α), which is an exceptional divisor in the sense that it is very rigidly embedded in X. The positive parts Z(α) generate a modified nef cone, and the pseudo-effective cone is shown to be locally polyhedral away from the modified nef cone, with extremal rays generated by exceptional divisors. We then treat the case of a surface and a hyper-Kähler manifold in some detail. Using the intersection form (respectively the Beauville-Bogomolov form), we characterize the modified nef cone and the exceptional divisors. The divisorial Zariski decomposition is orthogonal, and is thus a rational decomposition, which fact accounts for the usual existence statement of a Zariski decomposition on a projective surface, which is thus extended to the hyper-Kähler case. Finally, we explain how the divisorial Zariski decomposition of (the first Chern class of) a big line bundle on a projective manifold can be characterized in terms of the asymptotics of the linear series |kL| as k→∞.     

The Zariski Topology-Graph of Modules Over Commutative Rings

Let M be a module over a commutative ring, and let Spec(M) be the collection of all prime submodules of M. We topologize Spec(M) with Zariski topology, which is analogous to that for Spec(R), and for a nonempty subset T of Spec(M), we introduce a new graph G(τ _T ), called the Zariski topology-graph. This graph helps us to study the algebraic (resp. topological) properties of M (resp. Spec(M)) by using the graph theoretical tools.