Deformation equivalence of affine ruled surfaces

TL;DR

This paper provides a complete combinatorial classification of deformation equivalence among affine ruled surfaces, including the construction of comprehensive families and the potential for coarse moduli spaces in specific cases.

Contribution

It introduces a full combinatorial description of deformation equivalence for affine ruled surfaces and constructs complete families, advancing understanding of their moduli.

Findings

Complete combinatorial description of deformation equivalence

Construction of complete families of affine ruled surfaces

Existence of coarse moduli spaces in specific cases

Abstract

A smooth family $\varphi:\mathcal V\to S$ of surfaces will be called {\em completable} if there is a logarithmic deformation $(\bar {\mathcal V},{\mathcal D})$ over $S$ so that ${\mathcal V}=\bar{\mathcal V}\backslash {\mathcal D}$. Two smooth surfaces $V$ and $V'$ are said to be deformations of each other if there is a completable flat family ${\mathcal V}\to S$ of smooth surfaces over a connected base so that $V$ and $V'$ are fibers over suitable points $s,s'\in S$. This relation generates an equivalence relation called {\em deformation equivalence}. In this paper we give a complete combinatorial description of this relation in the case of affine ruled surfaces, which by definition are surfaces that admit an affine ruling $V\to B$ over an affine base with possibly degenerate fibers. In particular we construct complete families of such affine ruled surfaces. In a few particular cases we can also deduce the existence of a coarse moduli space.