Can the SL_2 character variety of a three-manifold be nonreduced? Let $M^3$ be a three-manifold and consider the representation variety and the character variety of $M$:
$$Y=\operatorname{Hom}(\pi_1(M^3),\operatorname{SL}(2,\mathbb C))$$
$$X=\operatorname{Hom}(\pi_1(M^3),\operatorname{SL}(2,\mathbb C))//\operatorname{SL}(2,\mathbb C)$$
I would like to think of these as schemes.  They are usually singular; in fact the trivial representation is almost always a singular point.
Are there any known examples where $Y$ (or $X$) are nonreduced as schemes?
I really want to know the answer for $\pi_1(M^3)$; examples with just any finitely presented group would be less interesting.
 A: There is actually an old (ca 1986) example of nonreduced $SL(2, {\mathbb C})$-representation scheme of a 3-manifold group. Take an oriented Seifert manifold $M$ which fibers over the $S^2(3,3,3)$ orbifold (sphere with three cone points of order 3). The fundamental group of the base-orbifold is von Dyck group with presentation
$$
\Gamma=\langle a, b, c | a^3, b^3, c^3, abc\rangle. 
$$ 
It is an old observation of Lubotzky and Magid (in their book "Representation varieties of finitely-generated groups") that the $SL(2, {\mathbb C})$-representation scheme of $\Gamma$ is nonreduced at a representation $\rho_0$ whose image is isomorphic to ${\mathbb Z}_3$ ($\rho_0$ sends all generators to an order $3$ element). Namely, $H^1(\Gamma, Ad \rho_0)$ is $1$-dimensional, while the representation $\rho_0$ s locally rigid. There is a nice geometric explanation of this phenomenon: Take a spherical equilateral triangle in $S^3=SU(2)$ contained in a great circle. Then this triangle is locally rigid but admits a nontrivial 1st order deformation in $S^3$. Now, $\rho_0$ lifts to an $SL(2, {\mathbb C})$-representation $\tilde\rho_0$ of the central extension $\pi=\pi_1(M)$ of the group $\Gamma$ (killing the center of $\pi$). This is your example. It is locally rigid but has 1-st order nontrivial infinitesimal deformations.  A drawback of this example is that the image has large centralizer. 
Below is a more difficult "universality" result:
Theorem 1. Let $X$ be an affine scheme over ${\mathbb Q}$ and $x\in X$ be a rational point. Then there exists an open subscheme $X'\subset X$ containing $x$, a natural number $N$, a closed 3-dimensional manifold $M$ with fundamental group $\pi$, a unitary representation $\rho: \pi\to SU(2)\subset SL(2, {\mathbb C})$ (whose image is dense in $SU(2)$) and an open subscheme 
$$
R'\subset Hom(\pi, SL(2, {\mathbb C}))
$$
containing $\rho$, so that $R'$ admits a regular etale covering over $X'\times SL(2, {\mathbb C})^N$ (with abelian group of deck transformations) and the covering sends $\rho$ to $x$. In particular, the centralizer of $\rho(\pi)$ is $\{\pm 1\}$. 
For instance, the analytic germ of the character scheme 
$$
Hom(\pi, SL(2, {\mathbb C}))//SL(2, {\mathbb C})$$ 
at $[\rho]$ could be isomorphic to the germ at $0$ of the scheme 
$$
\{x^{100}=0\} \times {\mathbb C}^{k} 
$$
(for some $k$). In particular, the character scheme of 3-manifold groups could be nonreduced at points of Zariski density. 
Proof of Theorem 1 could be now found here. 
The proof is a combination of my old work with Millson (see here) with the recent theorem of Panov and Petrunin, which deserves to be better known:
Theorem 2. For every finitely presented group $\Gamma$ there exists a  closed 3-dimensional orbifold $O$ so that the fundamental group of the underlying space of $O$ is isomorphic to $\Gamma$.  
It is a difficult open problem if Theorem 1 holds for 3-manifolds which are homology spheres. 
