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 ...

Definition 1.1. Let C be a category, and X, Y, Z objects of C. Fix also morphisms πX : X → Z, πY : Y → Z. Given this data, we say that an object P of C, together with morphisms p1 : P → X, p2 : P → Y is a fiber product of X with Y over Z if it satisfies the following universal property: For every object T ∈ Obj(C), and every pair of morphisms f : T → X, g : T → Y such that πX ◦ f = πY ◦ g, ther...

We present a Zariski pair in affine complex plane consisting of two line arrangements, each of which has six lines. In the seminal paper [3], Zariski started the study of the fundamental groups of the complements of plane algebraic curves. Among other things, he constructed a pair of plane curves with the same degree and the same local singularities, but non-isomorphic fundamental groups, which...