Rational cuspidal curves on del-Pezzo surfaces

TL;DR

This paper derives an explicit formula for counting rational cuspidal curves of a specified degree on del-Pezzo surfaces passing through generic points, using topological and Euler class techniques.

Contribution

It introduces a novel topological approach to compute the number of rational cuspidal curves on del-Pezzo surfaces via Euler class calculations.

Findings

Explicit enumeration formula derived

Topological method applied to degenerate locus

Counts of rational cuspidal curves obtained

Abstract

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.