Deformation and Unobstructedness of Determinantal Schemes

A closed subscheme X ⊂ P n X\subset \mathbb {P}^n is said to be determinantal if its homogeneous saturated ideal can be generated by the s × s s\times s minors of a homogeneous p × q p\times q matrix satisfying ( p − s + 1 ) ( q − s + 1 ) = n − dim ⁡ X (p-s+1)(q-s+1)=n - \dim X and it is said to be standard determinantal if, in addition, s = min ( p , q ) s=\min (p,q) . Given integers a 1 ≤ a 2 ≤ ⋯ ≤ a t + c − 1 a_1\le a_2\le \cdots \le a_{t+c-1} and b 1 ≤ b 2 ≤ ⋯ ≤ b t b_1\le b_2 \le \cdots \le b_t we consider t × ( t + c − 1 ) t\times (t+c-1) matrices A = ( f i j ) \mathcal {A}=(f_{ij}) with entries homogeneous forms of degree a j − b i a_j-b_i and we denote by W ( b _ ; a _ ; r ) ¯ \overline {W(\underline {b};\underline {a};r)} the closure of the locus W ( b _ ; a _ ; r ) ⊂ H i l b p ( t ) ( P n ) W(\underline {b};\underline {a};r)\subset Hilb^{p(t)}(\mathbb {P}^{n}) of determinantal schemes defined by the vanishing of the