We discuss the principle tools and results and state a few open problems concerning the classification and topology of plane sextics and trigonal curves in ruled surfaces.

Anabelian geometry is a grand synthesis of number theory, algebraic geometry, algebraic topology, and group theory. Parallel to the situation in algebraic topology where to a space, one attaches a fundamental group (after fixing a basepoint), in algebraic geometry one attaches to every algebraic variety X (and a geometric point x) its étale fundamental group πét 1 (X,x). Similar to the topologi...

HiX and H X = ⊕ H iX. We refer to [Spa66] for the details and basic properties of these constructions, summarizing the most relevant facts below. One sacrifices some geometric intuition in working with cohomology instead of homology, but one gains the advantage of an easily defined ring structure. If σ ∈ CkX is a singular simplex, let fiσ ∈ CiX be the restriction of σ to the front i-face of the...

In this paper we show that, after completing in the $I$-adic topology, Turaev cobracket on vector space freely generated by closed geodesics a smooth, complex algebraic curve $X$ with quasi-algebraic framing is morphism of mixed Hodge structure. We combine results previous Goldman bracket to construct torsors solutions Kashiwara–Vergne problem all genera. The so constructed form torsor under pr...

The classification of surfaces theorem was one of the earliest triumphs of algebraic topology. It states that any closed connected surface is homeomorphic to the sphere, the connected sum of n tori, or the connected sum of m projective planes. This paper begins by defining the geometric, topological, and algebraic tools necessary to understanding the theorem, then proceeds through the technical...

It appears that if dimcZ = a. dim X > 2, one may consider X as an algebraic variety too but in some new sense. One can generalize the conception of the abstract variety of A. Weil by substituting the etale topology of Grothendieck for the topology of Zariski. One gets the objects which M. Artin called "etale schemes" and the author called "minischemes". Later, M. Artin introduced the term "alge...