Generic determinantal schemes and the smoothability of determinantal schemes of codimension 2
The purpose of this paper is to prove a criterion for projectively Cohen- Macaulay-two-codimensional subschemes \(Y\) of \(\mathbb{P}_ k^ N\) (\(k\) algebraically closed of characteristic 0) to be smoothable. The considered schemes are determinantal, i.e. the homogeneous ideal \(I(Y)\), defining \(Y\), is generated by the \(t\times t\)-minors of a homogeneous \(m\times n\)-matrix \(M\) with \(\text{ht}(I(Y))= (m-t+1) (n-t+1)\). Determinantal schemes are equidimensional with a Cohen-Macaulay coordinate ring. It is easy to see that a closed subscheme \(Y\subset \mathbb{P}_ k^ N\) of equicodimension two with a Cohen-Macaulay coordinate ring is necessarily determinantal. The author determines first the codimension of the singular locus of \(Y\) (see propositions 1 and 2 in section 2.1) under certain conditions on the entries of \(M\). Then she proves the existence of a deformation \(f:X\to Z\) of \(Y\) (determinantal and of codimension 2) over an irreducible \(k\)-scheme \(Z\) (i.e. \(Y=f^{- 1}(z_ 0)\) over a closed point \(z_ 0\)); and \(Y\) is then smoothable iff \(2\leq N\leq 5\) (see theorem in section 2.2). This result generalizes a corresponding criterion of \textit{T. Sauer} [Math. Ann. 272, 83-90 (1985; Zbl 0546.14023)] for curves in \(\mathbb{P}_ k^ 3\).
scientific article; zbMATH DE number 572237