Classifying complete ℂ-subalgebras of ℂ[[t]]

2019/10/13 by Eloise Hamilton, Hamilton, Eloise
Mathematics · #14H10 (Primary) #14H20 #14Q05 (Secondary) #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG #msc:14H10 #msc:14H20 #msc:14Q05

paper · pdf · doi:10.48550/arxiv.1910.05753

20 pages

arxiv created 2019/10/13 · arxiv updated 2019/10/15

Abstract

We address the problem of classifying complete ℂ-subalgebras of ℂ[[t]]. A discrete invariant for this classification problem is the semigroup of orders of the elements in a given ℂ-subalgebra. Hence we can define the space R_Γ of all ℂ-subalgebras of ℂ[[t]] with semigroup Γ. After relating this space to the Zariski moduli space of curve singularities and to a moduli space of global singular curves, we prove that R_Γ is an affine variety by describing its defining equations in an ambient affine space in terms of an explicit algorithm. Moreover, we identify certain types of semigroups Γ for which R_Γ is always an affine space, and for general Γ we describe the stratification of R_Γ by embedding dimension. We also describe the natural map from R_Γ to the Zariski moduli space in some special cases. Explicit examples are provided throughout.

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