On the diameter of plane algebraic curves

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

Inflexion Points on Plane Algebraic Curves

In this thesis we will have a look at algebraic curves in the projective plane over an arbitrary algebraically closed field k. Using the resultant of polynomial rings over k we define intersection multiplicities and prove Bézout’s Theorem for effective divisors. We define singularities and inflexion points and count their number depending on the degree of the curve, using the Hessian of a curve.

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.