Special loci in moduli of marked curves
A \textit{special locus}, for a given compact orientable surface \(S\) with \(n\) unordered marked points, consists of a set of complex structures on \(S\) such that a fixed, finite--order diffeomorphism \(\phi\) of \(S\) is an automorphism of the corresponding algebraic surfaces. The author translates this notion in purely algebraic terms. If \(S\) is an algebraic curve of genus \(g\geq 2\), with \(n\) unordered marked points, and \(\alpha\) is a fixed automorphism of \(S\), the author defines the special locus of \(\alpha\) as the locus, in the moduli space of marked curves, of curves \(S'\) having an automorphism \(\alpha'\) such that the pair \((S',\alpha')\) is a flat deformation of \((S,\alpha)\). The author proves that this algebraic definition is equivalent to the topological one. The notion of special locus, consequently, can be extended also to curves defined over arbitrary algebraically closed fields, of any characteristic. For curves in positive characteristic, as Riemann--Hurwitz formulas are different, some new behaviour of special loci arises.
scientific article; zbMATH DE number 5509669
scientific article