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 .

We obtain an explicit formula for the number of rational cuspidal curves of a given degree on a del-Pezzo surface that pass through an appropriate number of generic points of the surface. This enumerative problem is expressed as an Euler class computation on the moduli space of curves. A topological method is employed in computing the contribution of the degenerate locus to this Euler class.

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 + 1)) generic points and having one node and one codimension k singularity, where k is at most 6. 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 .

No abstract

No abstract