The generic smoothness and matrix-deformation conjecture for determinantal schemes

Let $r\ge1$, $c>2-r$, and let $\mathcal{A}$ be a homogeneous $t\times(t+c-1)$ matrix with entries of degree $a_j-b_i$. Set $A=R/I_{t-r+1}(\mathcal{A})$ and suppose $\operatorname{Proj}(A)\in W(\underline{b};\underline{a};r)$, $\dim A\ge4$ for $c=1$, $\dim A\ge3$ for $c\ne0$, and $a_1>b_t$. The generic smoothness and matrix-deformation conjecture. Then $\overline{W(\underline{b};\underline{a};r)}$ is a generically smooth irreducible component of $\operatorname{Hilb}^{p_X}(\mathbb{P}^n)$, and every deformation of $A$ comes from deforming $\mathcal{A}$. This conjecture asserts both the expected Hilbert-scheme component property and completeness of matrix-induced deformations; the paper presents it as open and motivates it by examples and computational evidence.