On the normal sheaf of determinantal varieties

Abstract

Abstract Let X be a standard determinantal scheme X ⊂ ℙ n ${X\subset\mathbb{P}^{n}}$ of codimension c, i.e. a scheme defined by the maximal minors of a t × ( t + c - 1 ) $t\times(t+c-1)$ homogeneous polynomial matrix 𝒜 ${\mathcal{A}}$ . In this paper, we study the main features of its normal sheaf 𝒩 X ${{\mathcal{N}}_{X}}$ . We prove that under some mild restrictions: (1) there exists a line bundle ℒ ${\mathcal{L}}$ on X ∖ Sing ⁢ ( X ) ${X\setminus\mathrm{Sing}(X)}$ such that 𝒩 X ⊗ ℒ ${{\mathcal{N}}_{X}\otimes\mathcal{L}}$ is arithmetically Cohen–Macaulay and, even more, it is Ulrich whenever the entries of 𝒜 ${\mathcal{A}}$ are linear forms, (2) 𝒩 X ${{\mathcal{N}}_{X}}$ is simple (hence, indecomposable) and, finally, (3) 𝒩 X ${{\mathcal{N}}_{X}}$ is μ-(semi)stable provided the entries of 𝒜 ${\mathcal{A}}$ are linear forms.