An infinitesimal study of secant space divisors. Appendix by Trygve Johnsen
type: Item
instance of: scholarly article
title: An infinitesimal study of secant space divisors. Appendix by Trygve Johnsen
zbMATH Open document ID: 0890.14004
DOI: 10.1016/0022-4049(95)00098-4
published in: Journal of Pure and Applied Algebra
publication date: 1997-03-02T00:00:00Z
review text: The author studies \(e\)-secants to a smooth irreducible complete curve \(C\) defined over an algebraically closed field, considered as points in the \(e\)'th symmetric product \(C^{(e)}\) of \(C\). Such a secant is called an \((e-f-1)\)--space divisor for a linear system \(g_d^n\) if \(n\geq e-f\) and \(\text{dim}[g_d^n(D)] =n-e-f\). When \(g_d^n\) defines an embedding into a projective space \(\mathbb{P}^n\) we have that, when \(D\) is an \((e-f-1)\)--space divisor for \(g_d^n\), then the points of \(D\) span a subspace \(\langle D\rangle\) of \(\mathbb{P}^n\) of dimension \((e-f-1)\), and the intersection \(\langle D\rangle \cap C\) in \(\mathbb{P}^n\) will have length at least \(D\). In the article under review very precise information is given about the tangent space of the subscheme \(V_{e}^{e-f}\) of \(C^{(e)}\) consisting of \(e\)-secant \((e-f-1)\)--space divisors. The dimension is determined, thus answering a question by the author. Several of the results are similar to results of A. Laudal, in a slightly different setting. In an appendix T. Johnson gives a natural description for some of the results in terms of tangent spaces of natural subschemes of grassmannians.
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 14M15
Mathematics Subject Classification ID: 14H45
Mathematics Subject Classification ID: 14N05
zbMATH DE Number: 952352
zbMATH Keywords: linear system
zbMATH Keywords: symmetric product of curve
zbMATH Keywords: Zariski-tangent space
zbMATH Keywords: secant line divisor
zbMATH Keywords: Grassmannians
reviewed by: Dan Laksov
MaRDI profile type: Publication
full work available at URL: 0022-4049(95)00098-4
OpenAlex ID: W2004718524
cites work: Q3347989
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author: Marc Coppens
description: scientific article; zbMATH DE number 952352