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

TL;DR

This paper provides a complete description of the $ ext{SL}_2( ext{C})$-character scheme for small Seifert 3-manifolds, revealing conditions for its reducedness and the multiplicity of exceptional abelian characters.

Contribution

It extends previous work by fully characterizing the structure of the character scheme for all small Seifert 3-manifolds, including the behavior of exceptional abelian characters.

Findings

The character scheme is reduced iff the manifold has no exceptional abelian characters.

Exceptional abelian characters have multiplicity 2 in the scheme.

Complete classification of the character scheme structure for all small Seifert 3-manifolds.

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 $\mathcal{X}(M).$