Danielewski-Fieseler surfaces
Abstract
Description
We study a class of normal affine surfaces with additive group actions which contains in particular the Danielewski surfaces in $\ba^{3}$ given by the equations $x^{n}z=P(y)$, where $P$ is a nonconstant polynomial with simple roots. We call them Danielewski-Fieseler Surfaces. We reinterpret a construction of Fieseler \cite{Fie94} to show that these surfaces appear as the total spaces of certain torsors under a line bundle over a curve with an $r$-fold point. We classify Danielewski-Fieseler surfaces through labelled rooted trees attached to such a surface in a canonical way. Finally, we characterize those surfaces which have a trivial Makar-Limanov invariant in terms of the associated trees.
In this paper, we generalize the results on Danielewski surfaces to surfaces admitting certain A^1-fibration p:S-->X over the spectrum of a discrete valuation ring. We characterize among them the ones with a trivial Makar-Limanov invariant over an arbitrary algebraically closed field of caracteristic zero
In this paper, we generalize the results on Danielewski surfaces to surfaces admitting certain A^1-fibration p:S-->X over the spectrum of a discrete valuation ring. We characterize among them the ones with a trivial Makar-Limanov invariant over an arbitrary algebraically closed field of caracteristic zero