Special linear systems and syzygies
type: Item
instance of: scholarly article
title: Special linear systems and syzygies
zbMATH Open document ID: 1194.14019
DOI: 10.1007/BF03191185
published in: Collectanea Mathematica
publication date: 2010-03-04T00:00:00Z
full work available at URL: https://eudml.org/doc/42138
review text: The author considers linear systems \(\Lambda\) of hypersurfaces on a projective space \(\mathbb{P}^r(\mathbb{C})\) and the associated rational map \(\Phi:=\Phi_\Lambda :\mathbb{P}^r\to\mathbb{P}^\alpha .\) In Theorem 1 it is shown that the existence of linear syzygies for the homogeneuos ideal of the base locus of \(\Lambda\) has strong consequences for the fibers of \(\Phi.\) The result reminds a well-known fact according to which the \(K_d\) property of the base locus implies the linearity of the fibers of \(\Phi.\) The next main result, Theorem 3, concerns the special case of linear systems of quadrics. It states that, when the Gauss map of \(\Phi(\mathbb{P}^r)\) is degenerate, the inverse image of its non-trivial fibers is a linear space in \(\mathbb{P}^r.\) A number of interesting applications and examples are given.
reviewed by: Cristodor-Paul Ionescu
Mathematics Subject Classification ID: 14E05
Mathematics Subject Classification ID: 14M12
zbMATH DE Number: 5677586
zbMATH Keywords: quadrics
zbMATH Keywords: linear syzygies
zbMATH Keywords: rational map.
author: Alberto Alzati
MaRDI profile type: Publication
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OpenAlex ID: W2043666123
description: scientific article; zbMATH DE number 5677586