Degenerating curves and surfaces: first results

TL;DR

This paper investigates the degeneration of curves with specific singularities on a family of surfaces transitioning from smooth to reducible, identifying all limit curves with one cusp or node in the special fiber.

Contribution

It determines all irreducible components of the special fiber of the Universal Severi-Enriques variety for certain singularities, providing new insights into degenerations of singular curves.

Findings

Classifies limit curves with one cusp or node on the special fiber.

Identifies all irreducible components of the special fiber of the variety.

Provides a detailed description of degenerations of singular curves.

Abstract

Let $\mathcal S\to\mathbb A^1$ be a smooth family of surfaces whose general fibre is a smooth surface of $\mathbb P^3$ and whose special fibre has two smooth components, intersecting transversally along a smooth curve $R$. We consider the Universal Severi-Enriques variety $\mathcal V$ on $\mathcal S\to\mathbb A^1$. The general fibre of $\mathcal V$ is the variety of curves on $\mathcal S_t$ in the linear system $|\mathcal O_{\mathcal S_t}(n)|$ with $k$ cusps and $\delta$ nodes as singularities. Our problem is to find all irreducible components of the special fibre of $\mathcal V$. In this paper, we consider only the cases $(k,\delta)=(0,1)$ and $(k,\delta)=(1,0)$. In particular, we determine all singular curves on the special fibre of $\mathcal S$ which, counted with the right multiplicity, are a limit of 1-cuspidal curves on the general fibre of $\mathcal S$.