Families of low dimensional determinantal schemes

A scheme X⊂Pn of codimension c is called standard determinantal if its homogeneous saturated ideal can be generated by the t×t minors of a homogeneous t×(t+c−1) matrix (fij). Given integers a0≤a1≤⋯≤at+c−2 and b1≤⋯≤bt, we denote by the stratum of standard determinantal schemes where fij are homogeneous polynomials of degrees aj−bi and is the Hilbert scheme (if n−c>0, resp. the postulation Hilbert scheme if n−c=0).Focusing mainly on zero and one dimensional determinantal schemes we determine the codimension of in and we show that is generically smooth along under certain conditions. For zero dimensional schemes (only) we find a counterexample to the conjectured value of appearing in Kleppe and Miró-Roig (2005) [25].