Families of determinantal schemes

Given integers $a_0\le a_1\le \cdots \le a_{t+c-2}$ and $b_1\le \cdots \le b_t$, we denote by $W(\underline{b};\underline{a})\subset \textrm{Hilb}^p(\mathbb{P}^{n})$ the locus of good determinantal schemes $X\subset \mathbb{P}^{n}$ of codimension $c$ defined by the maximal minors of a $t\times (t+c-1)$ homogeneous matrix with entries homogeneous polynomials of degree $a_j-b_i$. The goal of this paper is to extend and complete the results given by the authors in an earlier paper and determine under weakened numerical assumptions the dimension of $W(\underline{b};\underline{a})$ as well as whether the closure of $W(\underline{b};\underline{a})$ is a generically smooth irreducible component of $\textrm{Hilb}^p(\mathbb{P}^{n})$.