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Enumeration of curves with one singular point
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In this paper we obtain an explicit formula for the number of degree d curves in two dimensional complex projective space, passing through (d(d+3)/2 -k) generic points and having a codimension k singularity, where k is at most 7. In the past, many of these numbers were computed using techniques from algebraic geometry. In this paper we use purely topological methods to count curves. Our main tool is a classical fact from differential topology: the number of zeros of a generic smooth section of a vector bundle V over M, counted with a sign, is the Euler class of V evaluated on the fundamental class of M.
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Cited by 1 Pith paper
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Counting curves in a linear system with upto eight singular points
The authors derive recursive Euler class formulas that enumerate curves with δ nodes and one fixed singularity for all δ+k ≤ 8, recovering prior results and producing new codimension eight numbers.
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