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Small Seifert 3-manifolds with non-reduced $\mathrm{SL}_2(\mathbb{C})$-character scheme

Renaud Detcherry

TL;DR

The paper determines when the SL_2(C)-character scheme ${\mathcal{X}}(M)$ of a small Seifert 3-manifold is non-reduced, showing that exceptional abelian characters cause non-reducedness with multiplicity 2. By leveraging the Heusener-Porti presentation of coordinate rings and a detailed tangent-space analysis at exceptional abelian characters, the author proves that ${\mathcal{X}}(M)$ is reduced precisely when no exceptional abelian character exists. The work provides an explicit presentation of ${\mathbb{C}}[ {\mathcal{X}}(M) ]$ via Chebyshev polynomials for Seifert data $(q_i/p_i)$ and demonstrates there are no order-2 deformations of exceptional abelian characters, solidifying the multiplicity-2 phenomenon. These results tie into skein-theoretic interpretations and show that skein-module dimensions over $\mathbb{Q}(A)$ with $A$ specialized to $-1$ reflect character-multiplicity data, aligning with prior DKS1, FKBT insights. Overall, the paper completes the description of character schemes for small Seifert manifolds and provides explicit, verifiable non-reduced examples.

Abstract

We complete the work started in previous work of the author and Kalfagianni and Sikora, and give a complete description of the $\mathrm{SL}_2(\mathbb{C})$-character scheme $\mathcal{X}(M)$ of all small Seifert $3$-manifolds $M$. We find that $\mathcal{X}(M)$ is reduced if and only if $M$ admits no exceptional abelian character, and that exceptional abelian character have multiplicity $2$ in $\mathcal{X}(M).$

Small Seifert 3-manifolds with non-reduced $\mathrm{SL}_2(\mathbb{C})$-character scheme

TL;DR

The paper determines when the SL_2(C)-character scheme of a small Seifert 3-manifold is non-reduced, showing that exceptional abelian characters cause non-reducedness with multiplicity 2. By leveraging the Heusener-Porti presentation of coordinate rings and a detailed tangent-space analysis at exceptional abelian characters, the author proves that is reduced precisely when no exceptional abelian character exists. The work provides an explicit presentation of via Chebyshev polynomials for Seifert data and demonstrates there are no order-2 deformations of exceptional abelian characters, solidifying the multiplicity-2 phenomenon. These results tie into skein-theoretic interpretations and show that skein-module dimensions over with specialized to reflect character-multiplicity data, aligning with prior DKS1, FKBT insights. Overall, the paper completes the description of character schemes for small Seifert manifolds and provides explicit, verifiable non-reduced examples.

Abstract

We complete the work started in previous work of the author and Kalfagianni and Sikora, and give a complete description of the -character scheme of all small Seifert -manifolds . We find that is reduced if and only if admits no exceptional abelian character, and that exceptional abelian character have multiplicity in
Paper Structure (9 sections, 13 theorems, 92 equations)

This paper contains 9 sections, 13 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Useful formulas
  3. Chebyshev polynomials
  4. Trace functions of powers
  5. Computation of ${\mathbb{C}}[{\mathcal{X}}(M)]$
  6. Presentation of ${\mathbb{C}}[{\mathcal{X}}(\Gamma)]$ from a presentation of $\Gamma$
  7. A presentation of ${\mathbb{C}}[{\mathcal{X}} (M)]$ for $M$ a small Seifert manifold
  8. The tangent space at exceptional abelian characters
  9. Order $2$ deformations

Key Result

Theorem 1.1

DKS2 Let $M=S^2(q_1/p_1,q_2,p_2,q_3/p_3)$ be a Seifert manifold and assume $e(M)\neq 0.$ Then where $p_i^+=\lceil \frac{p_i}{2} \rceil-1$ and $p_i^-=\lfloor \frac{p_i}{2} \rfloor$ and $x_M$ is the number of exceptional abelian characters. Moreover, a point $[\rho]$ in $X(M)$ is reduced, except possibly if $[\rho]$ is an exceptional abelian character.

Theorems & Definitions (22)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Proposition 3.1
  • ...and 12 more