Families of low dimensional determinantal schemes

AbstractA 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 Ws(b¯;a¯)⊂Hilb(Pn) the stratum of standard determinantal schemes where fij are homogeneous polynomials of degrees aj−bi and Hilb(Pn) 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 Ws(b¯;a¯) in Hilb(Pn) and we show that Hilb(Pn) is generically smooth along Ws(b¯;a¯) under certain conditions. For zero dimensional schemes (only) we find a counterexample to the conjectured value of dimWs(b¯;a¯) appearing in Kleppe and Miró-Roig (2005) [25]