×

Danielewski-Fieseler surfaces. (English) Zbl 1105.14083

Let \(k\) be a field of characteristic zero. Consider a pair (\(X=\text{Spec}\,(A)\), \(x_0=\text{div}\,(x)\)), where \(A\) is either a discrete valuation ring with uniformizing parameter \(x\) and residue field \(k\) or a polynomial ring \(k[x]\). A Danielewski-Fieseler surface with base \((A,x)\) (DFG, for short) is an integral affine \(X\)-scheme \(\pi:S\to X\) of finite type, restricting to a trivial line bundle over \(X\setminus\{x_0\}\), and such that \(\pi^{-1}(x_0)\) is non-empty and reduced, consisting of a disjoint union of curves isomorphic to affine line \(\mathbb{A}^1_k\).
In this paper, the author gives a combinatorial description of DFSs in terms of \((A,x)\)-labelled rooted trees, that is, pairs \(\gamma=(\Gamma,\sigma)\) consisting of a rooted tree \(\Gamma\) and a cochain \(\sigma\in A^n\), indexed by the terminal elements of \(\Gamma\), satisfying certain conditions with respect to the geometry of \(\Gamma\). Using a construction of K. H. Fieseler [Comment. Math. Helv. 69, 5–27 (1994; Zbl 0806.14033)], it is shown that DSFs appear as the total spaces of certain torsors under a line bundle over a curve with an \(r\)-fold point. A characterization of DFSs which have a trivial Makar-Limanov invariant in terms of the associated trees is obtained.

MSC:

14R25 Affine fibrations
14D06 Fibrations, degenerations in algebraic geometry
14J10 Families, moduli, classification: algebraic theory

Citations:

Zbl 0806.14033
PDFBibTeX XMLCite
Full Text: DOI arXiv