The Gale transform and multi-graded determinantal schemes
type: Item
instance of: scholarly article
title: The Gale transform and multi-graded determinantal schemes
zbMATH Open document ID: 1155.14034
author: Steven P. Diaz
author: Susan Marie Cooper
published in: Journal of Algebra
publication date: 2008-07-11T00:00:00Z
review text: Let \(U_i\) be a k-vector space of dimension \( s_i +1 \geq 2\) for \( i = 1, \ldots, k\) and \( k \geq 2\) and \(F\) a vector space of dimension \( n= \sum s_i\) and \( \varphi: F \rightarrow \otimes _{i=1}^k U_i \) a linear transformation and denote by \( \varphi_j: F \otimes k[ \bigoplus_{ i \neq j} U_i ](1,1,1,\dots 1) \) be the corresponding map of free modules. Let \(P_j = \prod_{i \neq j} P^{s_i}\) be the subscheme defined by the ideals of \((s_j +1) \times (s_j +1) \) minor determinants \( I_{s_j +1}( \varphi_j) \subset k[ \bigoplus_{i \neq j}U_i ]\). For any \((k-1)\)-tuple of positive integers \(a\) denote by \( \sigma_a( P_j)\) the Segre embedding given by divisors of type \(a\). The author's main theorem, proposition 3.1, states and proves the equivalence of the following conditions: (i) for all \( 1 \leq i \leq k\), \(\Gamma_i\) is \(0\)-dimensional and locally Gorenstein, \(\text{codim} (I_{s_i +1}( \varphi)) = n-s_i \) and \(\text{codim}( I_{s_i}( \varphi)) = n-s_i +1 \). (ii) \(\Gamma_1\) , \(\Gamma_2\) are \(0\)-dimensional. (iii) \(\text{codim}( I_{s_i + 1}( \varphi)) = n-s_1, \text{codim}( I_{s_1}( \varphi_1)) = n-s_1 + 1 \). (iv) \( \Gamma_1 \) is \(0\)-dimensional and locally Gorenstein. Moreover they prove in theorem 3.2 that if one of the equivalent statements of the above is true then all \(\Gamma_ i \) are isomorphic to each other and defining \( a_i = n -\sum_{l=i}^{k-1}(s_l -1) \) for \(s_k \geq 2 \) and \( b_i = \sum_{l = i+1}^{k-1} s_l \) for \( 1\leq i \leq k-2 \) and \( b_k = n -s_k -1 \). Define \( a= ( a_1, \ldots, a_{k-1}), b= ( b_1, \ldots, b_{k-2}, b_k)\) for \( 1 \leq i \leq k-2\) then the Gale transform of \( \sigma_a( \Gamma_k)\) is \( \sigma_b( \Gamma_{k-1})\). The authors' main motivation is to generalize the results of \textit{D. Eisenbud} and \textit{S. Popescu} [J. Algebra 230, No. 1, 127--173 (2000; Zbl 1060.14528)] to cover linear transformations from a vector space \(F\) to a tensor product of three or more vector space…
reviewed by: Isidro Nieto Baños
Mathematics Subject Classification ID: 14M12
Mathematics Subject Classification ID: 14N05
Mathematics Subject Classification ID: 14M05
zbMATH DE Number: 5300799
zbMATH Keywords: determinantal varieties
zbMATH Keywords: projective techniques
zbMATH Keywords: varieties defined by ring conditions
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2007.11.013
OpenAlex ID: W2062845726
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DOI: 10.1016/J.JALGEBRA.2007.11.013
description: scientific article; zbMATH DE number 5300799