The polar curves of plane algebraic curves in the Galois fields

Plane Algebraic Curves

We go over some of the basics of plane algebraic curves, which are planar curves described as the set of solutions of a polynomial in two variables. We study many basic notions, such as projective space, parametrization, and the intersection of two curves. We end with the group law on the cubic and search for torsion points.

Real Plane Algebraic Curves

We study real algebraic plane curves, at an elementary level, using as little algebra as possible. Both cases, affine and projective, are addressed. A real curve is infinite, finite or empty according to the fact that a minimal polynomial for the curve is indefinite, semi–definite nondefinite or definite. We present a discussion about isolated points. By means of the operator, these points can ...

Topology of Plane Algebraic Curves

Lifting Galois covers of algebraic curves

(1b) Question. For which pairs (C0,H) does a lift exist? Note that the lifting problem for C0 is formally smooth. However we will see that in general the lifting problem for (C0,H) can be obstructed; in some cases a lifting does not exist, in several cases ramification in R is needed to make a lifting possible. In order to have a positive answer to this question it suffices to consider the case...

Topology of Plane Algebraic Curves: the Algebraic Approach

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.