A series of Zariski pairs and four Zariski triplets were found by using lattice theory of K3 surfaces. There is a Zariski triplet of which one member is a deformation of another.

variety). For each i, let φi(x) be a C-dense quantifier-free Li-formula with parameters from K. Then we can find a K-definable rational function f : C → P which is non-constant, and has the property that the divisor f−1(0) is a sum of distinct points in ⋂n i=1 φi(K), with no multipliticities. (In particular, the support of the divisor contains no points from C(K)\C(K) and no points from C \ C.)...

Let $R$ be a commutative ring with identity and $M$ be a unitary$R$-module. The primary-like spectrum $Spec_L(M)$ is thecollection of all primary-like submodules $Q$ such that $M/Q$ is aprimeful $R$-module. Here, $M$ is defined to be RSP if $rad(Q)$ isa prime submodule for all $Qin Spec_L(M)$. This class containsthe family of multiplication modules properly. The purpose of thispaper is to intro...

We study analytic Zariski structures from the point of view of non-elementary model theory. We show how to associate an abstract elementary class with a one-dimensional analytic Zariski structure and prove that the class is stable, quasi-minimal and homogeneous over models. We also demonstrate how Hrushovski’s predimension arises in this general context as a natural geometric notion and use it ...

The notion of an analytic Zariski structure was introduced in [1] by the author and N.Peatfield in a form slightly different from the one presented here. Analytic Zariski generalises the previously known notion of a Zariski structure (see [2] for one-dimensional case and [3], [4] for the general definition) mainly by dropping the requirement of Noetherianity and weakening the assumptions on the...

Introduction Let G be a simply connected and connected semisimple group over an algebraically closed field k of characteristic p > 0. T ⊂ G is a maximal torus and R is the root system relative to T . X(T ) is the weight lattice. Let B ⊃ T be a Borel subgroup corresponding to the negative roots R− = R. Denote by Gr the r-th Frobenius kernel of G. The socle and radical structures of the cohomolog...

let $r$ be a commutative ring with identity and $m$ be a unitary$r$-module. the primary-like spectrum $spec_l(m)$ is thecollection of all primary-like submodules $q$ such that $m/q$ is aprimeful $r$-module. here, $m$ is defined to be rsp if $rad(q)$ isa prime submodule for all $qin spec_l(m)$. this class containsthe family of multiplication modules properly. the purpose of thispaper is to intro...

Finite Frobenius rings have been characterized as precisely those finite rings satisfying the MacWilliams extension property, by work of Wood. In the present note we offer a generalization of this remarkable result to the realm of Artinian rings. Namely, we prove that a left Artinian ring has the left MacWilliams property if and only if it is left pseudo-injective and its finitary left socle em...