Characterization of the singular locus by matrix rank

Characterize the singular locus of the determinantal quintic hypersurface \(\mathcal{Q}=\{\mathbf{p}\in\mathbb{A}^5\mid \det\mathbf{M}(\mathbf{p})=0\}\) at generic points by proving or disproving that it equals the locus where the symmetric matrix \(\mathbf{M}(\mathbf{p})\) has rank \(3\).

Background

The paper studies the determinantal quintic hypersurface $\mathcal{Q}$ defined by the vanishing of the determinant of a symmetric $5\times5$ matrix $\mathbf{M}$ whose entries are trace coordinates and boundary parameters for the $\mathrm{SL}_2(\mathbb{C})$ character variety of the five-punctured sphere. The authors derive an explicit description of the singular locus using the kernel vector $\mathbf{v}$ and identify both an essential component, where the matrix has corank at least two, and accidental components occurring when a boundary trace satisfies $b_a=\pm2$.

At a generic point, the authors state that the singular locus is two-dimensional and conjecturally coincides with the rank-three locus of $\mathbf{M}$. Establishing this equality would justify the subsequent local description of the generic singularity as an $A_1$ singularity with exceptional fiber $\mathbb{P}^1$.