Enumerative Geometry via Topological Computations

Abstract

Enumerative geometry is a rich and fascinating subject that has been studied extensively by algebraic geometers. In our thesis however, we approach this subject using methods from differential topology. The method comprises of two parts. The first part involves computing the Euler class of a vector bundle and evaluating it on the fundamental class of a manifold. This is straightforward. The second part involves perturbing a section and computing its contribution near the boundary. This is usually difficult. We have used this method to compute how many degree $d$ curves are there in $\C\P^2$ that pass through $\frac{d(d+3)}{2} -(\delta + m)$ points having $\delta$ nodes and one singularity of codimension $m$ provided $\delta+m\leq 7$. We also indicate how to extend this approach if $\delta+m$ is greater than $7$.