On the enumeration of complex plane curves with two singular points

Abstract

We study equi-singular strata of plane curves with two singular points of prescribed types. The method of the previous work [Kerner06] is generalized to this case. In particular we consider the enumerative problem for plane curves with two singular points of linear singularity types. First the problem for two ordinary multiple points of fixed multiplicities is solved. Then the enumeration for arbitrary linear types is reduced to the case of ordinary multiple points and to the understanding of "merging" of singular points. Many examples and numerical answers are given.

24 pages, the Mathematica file with explicit calculations is attached. Some typos removed. To appear in the International Mathematics Research Notices

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• A conjectural generating function for numbers of curves on surfaces
• Node polynomials for families: results and examples
• On the enumeration of complex plane curves with two singular points
• Enumeration of uni-singular algebraic hypersurfaces
• On the geometry of some strata of uni-singular curves
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Cited by in corpus (5)

• On the enumeration of complex plane curves with two singular points
• Counting planar curves in ${\mathbb{P}}^3$ with degenerate singularities
• Curve Counting à la Göttsche
• Counting curves on a general linear system with up to two singular points
• Counting singular curves with tangencies