Enumeration of curves with one singular point

Abstract

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.

arXiv comment61 pages; changed to larger version (this one) to facilitate reference regarding details

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Cited by in corpus (3)

• Counting planar curves in ${\mathbb{P}}^3$ with degenerate singularities
• Enumeration of Rational Cuspidal Curves via the WDVV equation
• Elliptic Gromov-Witten Invariants of Del-Pezzo Surfaces