Deforming reducible representations of surface and 2-orbifold groups

For a compact 2-orbifold with negative Euler characteristic ${\mathcal{O}}^2$, the variety of characters of ${\pi }_1({\mathcal{O}}^2)$ in ${\mathrm{S}\mathrm{L}}_n(\mathbb{R})$ is a non-singular manifold at $\mathbb{C}$-irreducible representations. In this paper we prove that when a $\mathbb{C}$-irreducible representation of ${\pi }_1({\mathcal{O}}^2)$ in ${\mathrm{S}\mathrm{L}}_n(\mathbb{R})$ is viewed in ${\mathrm{S}\mathrm{L}}_{n+1}(\mathbb{R})$, then the variety of characters is singular, and we describe the singularity.