Plane algebraic curves with prescribed singularities

TL;DR

This paper investigates the existence and properties of algebraic plane curves with prescribed singularities, focusing on conditions for their non-emptiness and smoothness, especially for curves with nodes and cusps.

Contribution

It provides a comprehensive overview of conditions for the existence and smoothness of equisingular families of plane curves with arbitrary singularities, emphasizing asymptotic behavior.

Findings

Necessary and sufficient conditions for non-emptiness of ESFs.

Conditions for T-smoothness of ESFs.

Asymptotic equivalence of conditions for large degree.

Abstract

We report on the problem of the existence of complex and real algebraic curves in the plane with prescribed singularities up to analytic and topological equivalence. The question is whether, for a given positive integer $d$ and a finite number of given analytic or topological singularity types, there exist a plane (irreducible) curve of degree $d$ having singular points of the given type as its only singularities. The set of all such curves is a quasi-projective variety, which we call an equisingular family (ESF). We describe, in terms of numerical invariants of the curves and their singularities, the state of the art concerning necessary and sufficient conditions for the non-emptiness and $T$-smoothness (i.e., smooth of expected dimension) of the corresponding ESF. The considered singularities can be arbitrary, but we spend special attention to plane curves with nodes and cusps, the most studied case, where still no complete answer is known in general. An important result is, however, that the necessary and the sufficient conditions show the same asymptotics for $T$-smooth equisingular families if the degree goes to infinity.