Let V be a valuation domain of rank one with quotient field K. We study the set extensions to rational functions K(X) induced by pseudo-convergent sequences K from topological point view, endowing this either Zariski or constructible topology. In particular, we consider two subspaces prescribed breadth pseudo-limit. give some necessary conditions for space metrizable (under topology) in terms v...

<abstract><p>Let $ R be a G graded commutative ring and M $-graded $-module. The set of all second submodules is denoted by Spec_G^s(M), it called the spectrum $. We discuss rings with Noetherian prime spectrum. In addition, we introduce notion Zariski socle explore their properties. also investigate Spec^s_G(M) topology from viewpoint being space.</p></abstract>

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...

The strength of a homogeneous polynomial (or form) is the smallest length an additive decomposition expressing it whose summands are reducible forms. Using functors, we show that set forms with bounded not always Zariski-closed. More specifically, if ground field algebraically closed, prove quartics ≤3 Zariski-closed for large number variables.

Every Grassmannian, in its Plücker embedding, is defined by quadratic polynomials. We prove a vast, qualitative, generalisation of this fact to Plücker varieties, which are families of varieties in exterior powers of vector spaces that, like Grassmannians, are functorial in the vector space and behave well under duals. A special case of our result says that for each fixed natural number k, the ...

Using the Serre construction, we give examples of stable rank 2 vector bundles on principally polarized abelian threefolds (X,Θ) with Picard number 1. The Chern classes (c1, c2) of these examples realize roughly one half of the classes that are a priori allowed by the Bogomolov inequality and Riemann-Roch (the latter gives a certain divisibility condition). In the case of even c1, we study defo...

with dim Vi = si. The space Mord(P 1,F(n; s)) of morphisms from P1 to F(n; s) of multidegree d = (d1, ..., dl) can be viewed as the space of successive quotients of VP1 of vector bundles of rank ri and degree di, where ri := n − si and VP1 := V ⊗ OP1 is a trivial rank n vector bundle over P1. Its compactification, the hyperquot scheme which we denote HQd = HQd(F(n; s)), parametrizes flat famili...