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Cohen-Macaulay Type via Lattice Homology and the Motivic Poincaré Series

Alex Hof, András Némethi

TL;DR

This work develops a lattice-homology framework to classify reduced complex-analytic curve germs by Cohen-Macaulay type, connecting indecomposable MCM modules with discrete invariants and spectral sequences. The authors establish precise criteria for finite and tame CM types in terms of the weight function $w_0^C$ and the lattice homology of the germ and its subcurves, and they reformulate these characterizations via motivic Poincaré series. The key contributions include a complete lattice-homological description of finite CM type (via dominance by ADE germs) and tame CM type (via overrings and spectral data), plus a bridge to the motivic Poincaré series that encodes these properties in a univariate series. The results offer a practical, topology-flavored toolkit for discerning CM type from discrete invariants, potentially simplifying computations and enabling new insights in plane curve singularities and their moduli.

Abstract

We give results on reduced complex-analytic curve germs which relate their indecomposable maximal Cohen-Macaulay (MCM) modules to their lattice homology groups and related invariants, thereby providing a connection between the algebraic theory of MCM modules and techniques arising from low-dimensional topology. In particular, we characterize the germs $(C, o)$ of finite Cohen-Macaulay type in terms of the lattice homology $\mathbb{H}_*(C, o)$, and those of tame type in terms of the lattice homologies and associated spectral sequences of $(C, o)$ and its subcurves, including the distinction between germs of finite and infinite growth. As a consequence of these results, we obtain corresponding characterizations of a germ's Cohen-Macaulay type in terms of the motivic Poincaré series.

Cohen-Macaulay Type via Lattice Homology and the Motivic Poincaré Series

TL;DR

This work develops a lattice-homology framework to classify reduced complex-analytic curve germs by Cohen-Macaulay type, connecting indecomposable MCM modules with discrete invariants and spectral sequences. The authors establish precise criteria for finite and tame CM types in terms of the weight function and the lattice homology of the germ and its subcurves, and they reformulate these characterizations via motivic Poincaré series. The key contributions include a complete lattice-homological description of finite CM type (via dominance by ADE germs) and tame CM type (via overrings and spectral data), plus a bridge to the motivic Poincaré series that encodes these properties in a univariate series. The results offer a practical, topology-flavored toolkit for discerning CM type from discrete invariants, potentially simplifying computations and enabling new insights in plane curve singularities and their moduli.

Abstract

We give results on reduced complex-analytic curve germs which relate their indecomposable maximal Cohen-Macaulay (MCM) modules to their lattice homology groups and related invariants, thereby providing a connection between the algebraic theory of MCM modules and techniques arising from low-dimensional topology. In particular, we characterize the germs of finite Cohen-Macaulay type in terms of the lattice homology , and those of tame type in terms of the lattice homologies and associated spectral sequences of and its subcurves, including the distinction between germs of finite and infinite growth. As a consequence of these results, we obtain corresponding characterizations of a germ's Cohen-Macaulay type in terms of the motivic Poincaré series.

Paper Structure

This paper contains 47 sections, 40 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Preliminaries. Invariants of reduced curve germs
  3. Normalization, valuations, and the Hilbert function (AgNeIVAgNeVKNS2)
  4. Connection with the semigroup of values (delaMata87delaMataGarcia)
  5. The conductor and the $\delta$-invariant
  6. Subcurves and their invariants (GorNem2015)
  7. Preliminaries. Relationships between reduced curve germs
  8. (Classical) classification of reduced plane curve germs
  9. Birational maps and dominance
  10. Preliminaries. Cohen-Macaulay properties
  11. Cohen-Macaulay modules and local rings
  12. Local rings of finite CM type (D1DG1DG2DG4GKKnLWYo)
  13. Local rings of infinite CM type
  14. Conditions for tameness
  15. Preliminaries. Lattice homology of reduced curve germs
  16. ...and 32 more sections

Key Result

Theorem 1

Let $(C, o)$ be a reduced complex-analytic curve germ. Then $(C, o)$ is of finite Cohen-Macaulay type if and only if $\min w_0^C \ge -1$. More specifically, we have the following equivalences:

Theorems & Definitions (95)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Remark 1.0.1
  • Definition 2.1.1
  • Proposition 2.1.2
  • Definition 2.1.3
  • Definition 2.2.1
  • Lemma 2.2.5
  • ...and 85 more