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Anabelian properties of tame fundamental groups of singular curves

Takahiro Murotani

TL;DR

This work investigates how the tame fundamental group, particularly its geometrically pro-$\Sigma$ part, encodes the geometry of singular curves. Building on Das's result that the tame fundamental group of a singular curve over an algebraically closed field decomposes as a free product of the normalizations’ tame groups and a free pro-$\Sigma$ factor, the authors develop group-theoretic reconstruction methods to recover the normalization from tame fundamental group data across several field settings. They show that, under suitable hypotheses (infinite $\Sigma$, positive-genus components, and non-isotriviality conditions, among others), isomorphisms of the appropriate tame group data force isomorphisms of the normalizations, extending anabelian results from smooth to singular curves. The methods hinge on recovering the decomposition groups of irreducible components from $\Delta$ or $\Pi$ and then applying known anabelian results for smooth curves, with diverse applications to finite fields, generalized sub-$p$-adic fields, and related settings. This advances the program of understanding how geometric structure of singular curves is reflected in their tamely ramified Galois-theoretic invariants, illuminating when the normalization is recoverable from group data alone.

Abstract

In anabelian geometry, we consider to what extent the étale or tame fundamental groups of schemes reflect geometric properties of the schemes. Although there are many known results (mainly for smooth curves) in this area, general singular curves have rarely been treated. One reason is that we cannot determine the isomorphism classes of singular curves themselves from their étale or tame fundamental groups in general. On the other hand, Das proved that the structure of the tame fundamental group of a singular curve over an algebraically closed field is determined by its normalization and an invariant relating to its singularities. In the present paper, by using this and known anabelian results, we determine the isomorphism classes of the normalizations of singular curves over various fields under certain conditions.

Anabelian properties of tame fundamental groups of singular curves

TL;DR

This work investigates how the tame fundamental group, particularly its geometrically pro- part, encodes the geometry of singular curves. Building on Das's result that the tame fundamental group of a singular curve over an algebraically closed field decomposes as a free product of the normalizations’ tame groups and a free pro- factor, the authors develop group-theoretic reconstruction methods to recover the normalization from tame fundamental group data across several field settings. They show that, under suitable hypotheses (infinite , positive-genus components, and non-isotriviality conditions, among others), isomorphisms of the appropriate tame group data force isomorphisms of the normalizations, extending anabelian results from smooth to singular curves. The methods hinge on recovering the decomposition groups of irreducible components from or and then applying known anabelian results for smooth curves, with diverse applications to finite fields, generalized sub--adic fields, and related settings. This advances the program of understanding how geometric structure of singular curves is reflected in their tamely ramified Galois-theoretic invariants, illuminating when the normalization is recoverable from group data alone.

Abstract

In anabelian geometry, we consider to what extent the étale or tame fundamental groups of schemes reflect geometric properties of the schemes. Although there are many known results (mainly for smooth curves) in this area, general singular curves have rarely been treated. One reason is that we cannot determine the isomorphism classes of singular curves themselves from their étale or tame fundamental groups in general. On the other hand, Das proved that the structure of the tame fundamental group of a singular curve over an algebraically closed field is determined by its normalization and an invariant relating to its singularities. In the present paper, by using this and known anabelian results, we determine the isomorphism classes of the normalizations of singular curves over various fields under certain conditions.

Paper Structure

This paper contains 5 sections, 18 theorems, 41 equations.

Table of Contents

  1. Notations and conventions
  2. Fundamental groups and irreducible components of projective singular curves over separably closed fields
  3. Anabelian results for projective singular curves over various fields
  4. Tame fundamental groups of non-projective singular curves over finite fields
  5. Fundamental groups of non-projective singular curves over generalized sub-$p$-adic fields

Key Result

Theorem A

Suppose that, for $\Box\in\{\circ, \bullet\}$, each irreducible component of the normalization of $(X_\Box)_{k^\mathrm{s}_\Box}$ is of genus at least two, and that $\Sigma_\Box$ is an infinite set. Suppose, moreover, that there exists an isomorphism of profinite groups $\phi:\Pi_\circ\stackrel{\sim}

Theorems & Definitions (29)

  • Theorem A: cf. Remark \ref{['sigma']} and Theorem \ref{['projective']}
  • Theorem B: cf. Theorems \ref{['mainfin']} and \ref{['mainsubp']}
  • Remark 1.1
  • Proposition 1.2
  • Proposition 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Proposition 2.1
  • Remark 2.2
  • Definition 2.3
  • ...and 19 more