Enumeration of curves with one singular point

Abstract

In this paper we obtain an explicit formula for the number of curves in P 2 , of degree d, 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 .

Enumeration of curves with two singular points

“…is a generic smooth section. In [1], we constructed a section Ψ PX k of an appropriate vector bundle…”

“…are the contributions of the section π * 2 Ψ PX k ⊕ Q to the Euler class from the points of A 1 • B 1 and ∆B 2 respectively. The number C A 1 •B 1 (π * 2 Ψ PX k ⊕ Q) was computed in [1]. The main content of this paper is to compute C ∆B 2 (π * 2 Ψ PX k ⊕ Q).…”

“…Therefore, we get a recursive formula for N (A 1 PX k , n, m) in terms of N (A 1 PX l , n ′ , m ′ ) and N (PX k+1 , n, m). The main result of [1] was to find an explicit formula for N (PX k+1 , n, m). Using this and iterations, we get an explicit formula for N (A 1 PX k , n, m).…”

“…In this paper, we extend the methods applied in [1] to enumerate curves with two singular points. More precisely, we obtain an explicit answer for the following question: Question 1.2.…”

“…In [1], we obtained an explicit formula for N (X k , n), the number of degree d curves in P 2 that pass through δ d − (k + n) generic points having one singularity of type X k at the intersection of n generic lines. We extend the methods applied in [1] to obtain an explicit formula for N (A 1 X k , n).…”