Semi-group Conditions for Affine Algebraic Plane Curves with More than One Place at Infinity

Construction of Affine Plane Curves with One Place at Infinity

Rational curves with one place at infinity

Let K be an algebraically closed field of characteristic zero. Given a polynomial f(x, y) ∈ K[x, y] with one place at infinity, we prove that either f is equivalent to a coordinate, or the family (fλ)λ∈K has at most two rational elements. When (fλ)λ∈K has two rational elements, we give a description of the singularities of these two elements.

On irregular links at infinity of algebraic plane curves

We give two proofs of a conjecture of Neumann [N1] that a reduced algebraic plane curve is regular at infinity if and only if its link at infinity is a regular toral link. This conjecture has also been proved by Ha H. V. [H] using Lojasiewicz numbers at infinity. Our first proof uses the polar invariant and the second proof uses linear systems of plane curve singularities. The second approach a...

Curves Having One Place at Infinity and Linear Systems on Rational Surfaces

Denoting by Ld(m0, m1, . . . , mr) the linear system of plane curves passing through r + 1 generic points p0, p1, . . . , pr of the projective plane with multiplicity mi (or larger) at each pi, we prove the Harbourne-Hirschowitz Conjecture for linear systems Ld(m0, m1, . . . , mr) determined by a wide family of systems of multiplicities m = (mi) r i=0 and arbitrary degree d. Moreover, we provid...

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.