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...

Introduction The existence of singular algebraic curves with given invariants and given set of singular-ities and, at the same time, the study of (equisingular) families of such curves is a very old, but still attractive and widely open, problem. Already, at the beginning of the 20th century, the foundations were made in the works of Pl ucker, Severi, Segre and Zariski. In the sequel the theory...

The work of Dixmier in 1977 and Moeglin in 1980 show us that for a prime ideal P in the universal enveloping algebra of a complex finite-dimensional Lie algebra the properties of being primitive, rational and locally closed in the Zariski topology are all equivalent. This equivalence is referred to as the Dixmier-Moeglin equivalence. In this thesis we will study skew Laurent polynomial rings of...

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.

We factor the virtual Poincaré polynomial of every homogeneous space G/H , where G is a complex connected linear algebraic group and H is an algebraic subgroup, as t(t − 1)QG/H(t ) for a polynomial QG/H with non-negative integer coefficients. Moreover, we show that QG/H(t ) divides the virtual Poincaré polynomial of every regular embedding of G/H , if H is connected. Introduction and statement ...

We adapt the study of hyperspaces and function spaces from classical topology to digital topology. define graphs, some their relationships graphical properties.