Classifying complete $$\mathbb {C}$$ C -subalgebras of $$\mathbb {C}[[t]]$$ C [ [ t ] ]
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© The Author(s) 2019
Eloise Hamilton
Classifying complete C-subalgebras of C[[t]] Received: 16 March 2017 / Accepted: 8 November 2019 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 of all C-subalgebras of C[[t]] with semigroup . 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 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 for which R is always an affine space, and for general we describe the stratification of R by embedding dimension.We also describe the natural map from R to the Zariski moduli space in some special cases. Explicit examples are provided throughout.
Contents Classifying complete C-subalgebras of C[[t]] Abstract. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. 1 Geometric relevance of the space R . . . . . . . . . . . . . . . . 1.1. 1.1 R and the Zariski moduli space M of curve singularities 1.2. 1.2 R and the moduli space of global singular curves . . . . . 2. 2 R is an affine variety . . . . . . . . . . . . . . . . . . . . . . . . 3. 3 Computing R : examples . . . . . . . . . . . . . . . . . . . . . . 4. 4 Properties of R . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. 4.1 When is R an affine space? . . . . . . . . . . . . . . . . . 4.2. 4.2 Stratification of R by embedding dimension . . . . . . . . 4.3. 4.3 The map from M to R . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Introduction In this paper we consider the following algebraic problem: the classification of complete C-subalgebras of the ring of formal power series in one variable C[[t]]. E. Hamilton (B): University of Oxford, Oxford, UK e-mail: [email protected] Mathematics Subject Classification: 14H10 (primary) · 14H20
https://doi.org/10.1007/s00229-019-01162-5
E. Hamilton
As is often the case for classification problems in algebraic geometry, the problem can be broken down into two steps. First, we search for a discrete invariant which provides an initial coarse classification of the objects. Then, for each fixed value of the invariant, we search for an algebraic variety which parametrises all objects with this given value. A discrete invariant for our problem is given by a semigroup in N, obtained by taking the orders of elements of a given C-subalgebra of C[[t]]. Definition 0.1. Let R be a C-subalgebra of C[[t]]. The semigroup of R is the set R ⊆ N defined by R := {n ∈ N | ∃ f ∈ R ∗ with
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