Classifying complete mathbb {C}-subalgebras of mathbb {C}[[t

Abstract

We address the problem of classifying complete mathbb {C}-subalgebras of mathbb {C}[[t]]. A discrete invariant for this classification problem is the semigroup of orders of the elements in a given mathbb {C}-subalgebra. Hence we can define the space mathcal {R}_{Gamma } of all mathbb {C}-subalgebras of mathbb {C}[[t]] with semigroup Gamma . After relating this space to the Zariski moduli space of curve singularities and to a moduli space of global singular curves, we prove that mathcal {R}_{Gamma } 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 Gamma for which mathcal {R}_{Gamma } is always an affine space, and for general Gamma we describe the stratification of mathcal {R}_{Gamma } by embedding dimension. We also describe the natural map from mathcal {R}_{Gamma } to the Zariski moduli space in some special cases. Explicit examples are provided throughout.