Families of artinian and low dimensional determinantal rings

Let $\mathrm{GradAlg}(H)$ be the scheme parameterizing graded quotients of $R=k[x_0,\dots ,x_n]$ with Hilbert function H (it is a subscheme of the Hilbert scheme of ${\mathbb{P}}^n$ if we restrict to quotients of positive dimension, see definition below). A graded quotient $A=R/I$ of codimension c is called standard determinantal if the ideal I can be generated by the $t\times t$ minors of a homogeneous $t\times (t+c?1)$ matrix $(f_{ij})$. Given integers $a_0\le a_1\le ...\le a_{t+c?2}$ and $b_1\le ...\le b_t$, we denote by $W_s(\underset{\_}{b};\underset{\_}{a})?\mathrm{GradAlg}(H)$ the stratum of determinantal rings where $f_{ij}\in R$ are homogeneous of degrees $a_j?b_i$.In this paper we extend previous results on the dimension and codimension of $W_s(\underset{\_}{b};\underset{\_}{a})$ in $\mathrm{GradAlg}(H)$ to artinian determinantal rings, and we show that $\mathrm{GradAlg}(H)$ is generically smooth along $W_s(\underset{\_}{b};\underset{\_}{a})$ under some assumptions. For zero and one dimensional determinantal schemes we generalize earlier results on these questions. As a consequence we get that the general element of a component W of the Hilbert scheme of ${\mathbb{P}}^n$ is glicci provided W contains a standard determinantal scheme satisfying some conditions. We also show how certain ghost terms disappear under deformation while other ghost terms remain and are present in the minimal resolution of a general element of $\mathrm{GradAlg}(H)$.