On the existence of plane curves with imposed multiple points

On the existence of plane curves with imposed multiple points

We prove that a plane curve of degree d with r points of multiplicity m must have d ≥ m (r − 1) r−1

On the Existence of Plane Curves with Prescribed Singularities

On the Dimension of Linear Systems of Plane Curves with General Multiple Base Points

Fixing n general points pi in the plane, what is the dimension of the space of plane curves of degree d having multiplicity mi at pi for each i? In this article we propose an approach to attack this problem, and demonstrate it by successfully computing this dimension for all n and for mi constant, at most 3. Our approach is based on an analysis of the corresponding linear system on a degenerati...

Constructions of Plane Curves with Many Points

In this paper we investigate some plane curves with many points over Q, finite fields and cyclotomic fields. In a previous paper [4] the first two authors constructed a sequence of absolutely irreducible polynomials Pd(x, y) ∈ Z[x, y] of degree d having low height and many integral solutions to Pd(x, y) = 0. (The definition of these polynomials will be recalled in §4.) Here we construct further...

Singular Points of Plane Curves

ly isomorphic to (C×)r−1 × (C), and hence also to (S1)r−1 × (R), where r = |J | is the number of branches and k = δ(C)− r+1 = 1 2 (μ(C) + 1 − r). The construction of the Jacobian variety J(C̃) of the non-singular curve C̃ in the large is standard in algebraic geometry. There is also a notion of Jacobian of a singular curve C , defined e.g. in [85], which, like the other, is an abelian group. Ther...