Rational cuspidal curves on del-Pezzo surfaces

TL;DR

The paper develops a topological Euler-class method to count rational cuspidal curves of a fixed degree $\beta$ on complex del-Pezzo surfaces $X$, through $\delta_\beta-1$ generic points. It constructs a precise Euler-class setup on the moduli space $\overline{\mathcal{M}}_{0,\delta_\beta}(X,\beta)$, analyzes boundary degenerations via a detailed gluing construction, and reduces the cusp count to intersection numbers of tautological classes, yielding an explicit formula for $C_\beta$ in terms of the genus-zero counts $N_\beta$ and splittings $\beta_1+\beta_2=\beta$. The main result holds under the hypothesis $N_{\beta-3L}>0$ and matches known algebro-geometric results in special cases, while providing a framework that extends to higher-dimensional manifolds and other singularities. The approach combines explicit boundary analysis, transversality arguments, and a constructive neighborhood/gluing analysis to enable enumerative computations of cuspidal rational curves on del-Pezzo surfaces and beyond.

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.

Theorems & Definitions (18)

• Theorem 1.2
• Lemma 4.1
• proof
• proof : Proof of \ref{['c2']}
• proof : Proof of \ref{['c1L']}
• proof : Proof of \ref{['c1_sq']}
• Lemma 5.1
• proof
• Corollary 5.2
• proof