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

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 $\mathcal{X}(M).$