Plane algebraic curves with prescribed singularities
Gert-Martin Greuel, Eugenii Shustin

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 and a finite number of given analytic or topological singularity types, there exist a plane (irreducible) curve of degree 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 -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…
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Taxonomy
TopicsCommutative Algebra and Its Applications · Algebraic Geometry and Number Theory · Polynomial and algebraic computation
