2013/05/23 by Flenner, Hubert, Kaliman, Shulim, Zaidenberg, Mikhail
#14D22 #14J10 #14R05 #14R25 #Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1305.5366
A smooth family φ:\mathcal V→ S of surfaces will be called \em completable if there is a logarithmic deformation ( \mathcal V,\mathcal D) over S so that \mathcal V=\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→ S of smooth surfaces over a connected base so that V and V' are fibers over suitable points s,s'∈ 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→ 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.