Multiple points of algebraic curves

Tropical and algebraic curves with multiple points

Patchworking theorems serve as a basic element of the correspondence between tropical and algebraic curves, which is a core of the tropical enumerative geometry. We present a new version of a patchworking theorem which relates plane tropical curves with complex and real algebraic curves having prescribed multiple points. It can be used to compute Welschinger invariants of non-toric Del Pezzo su...

Abelian Points on Algebraic Curves

We study the question of whether algebraic curves of a given genus g defined over a field K must have points rational over the maximal abelian extension K of K. We give: (i) an explicit family of diagonal plane cubic curves without Q-points, (ii) for every number field K, a genus one curve C/Q with no K -points, and (iii) for every g ≥ 4 an algebraic curve C/Q of genus g with no Q-points. In an...

Acquisition of Rational Points on 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...

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.