On the Shape of projective plane algebraic Curves

Solvable Points on Projective Algebraic Curves

We examine the problem of finding rational points defined over solvable extensions on algebraic curves defined over general fields. We construct non-singular, geometrically irreducible projective curves without solvable points of genus g when g is at least 40 over fields of arbitrary characteristic. We prove that every smooth, geometrically irreducible projective curve of genus 0, 2, 3 or 4 def...

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 ...

Asymptotes, Cubic Curves, and the Projective Plane

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Topology of Plane Algebraic Curves