Generic determinantal schemes and the smoothability of determinantal schemes of codimension 2
type: Item
instance of: scholarly article
title: Generic determinantal schemes and the smoothability of determinantal schemes of codimension 2
zbMATH Open document ID: 0795.14025
DOI: 10.1007/BF02567711
author: Frauke Steffen
published in: Manuscripta Mathematica
publication date: 1994-09-15T00:00:00Z
full work available at URL: https://eudml.org/doc/155941
review text: 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 \(\mathbb{P}_ k^ 3\).
Mathematics Subject Classification ID: 14M12
Mathematics Subject Classification ID: 13C40
Mathematics Subject Classification ID: 13D10
zbMATH DE Number: 572237
zbMATH Keywords: determinantal schemes
zbMATH Keywords: smoothability
reviewed by: Manfred Herrmann
MaRDI profile type: Publication
cites work: Q3813899
cites work: Q4143433
cites work: Rings of invariants of reductive groups acting on regular rings are Cohen-Macaulay
cites work: Deformations of Cohen-Macaulay Schemes of Codimension 2 and Non-Singular Deformations of Space Curves
description: scientific article; zbMATH DE number 572237