On the space of projective curves of maximal regularity
type: Item
instance of: scholarly article
title: On the space of projective curves of maximal regularity
zbMATH Open document ID: 1358.14025
author: Kiryong Chung
author: Wanseok Lee
author: Euisung Park
published in: Manuscripta Mathematica
publication date: 2016-12-20T00:00:00Z
review text: Let \(r,d\) be integers such that \(r\geq 3\), \(d\geq r+2\). Let \(C\subseteq {\mathbb P}^r\) be a nondegenerate, irreducible, projective curve of degree \(d\); \(C\) is called \(m\)-regular if \(H^i({\mathbb P}^r, {\mathcal I}_C(m-i))=0\) for \(i\geq 1\). The \textit{regularity} \(\text{reg}(C)\) of \(C\) is the least integer \(m\) such that \(C\) is \(m\)-regular. We have \(\text{reg}(C)\leq d-r+2\) [\textit{L. Gruson} et al., Invent. Math. 72, 491--506 (1983; Zbl 0565.14014)], and if equality holds, \(C\) is named of \textit{maximal regularity}. In this case, \(C\) is a nonsingular rational curve admiting a \((d-r+2)\)-secant line, the so-called \textit{extremal secant line} [loc. cit.]. In this paper the authors are interested in spaces that parametrize projective curves with a fixed regularity condition; in particular, they show that \(R=R^{d-r+2}_{r,d}\), the set of all maximal regularity curves in \({\mathbb P}^r\) of degree \(d\), is an irreducible variety of dimension \(3d+r^2-r-1\) (cf. Lemma 2.4 in [\textit{T. Johnsen} and \textit{S. L. Kleiman}, Commun. Algebra 24, No. 8, 2721--2753 (1996; Zbl 0860.14038)]. Moreover, as the automorphism group \(A=\mathrm{PGL}(r+1)\) of \({\mathbb P}^r\) acts in a natural way on \(R\), we can define in particular the stabilizer group \(A_C\) for each \(C\in R\); they show that \(A_C\) is finite and \(C\) cuts out the extremal secant line in at least 4 points.
reviewed by: Fernando Torres
Mathematics Subject Classification ID: 14H45
Mathematics Subject Classification ID: 14D23
Mathematics Subject Classification ID: 51N35
zbMATH DE Number: 6666256
zbMATH Keywords: rational curve
zbMATH Keywords: \(m\)-regular curve
zbMATH Keywords: regularity of a curve
zbMATH Keywords: extremal secant line
describes a project that uses: SINGULAR
MaRDI profile type: Publication
full work available at URL: s00229-016-0844-0
OpenAlex ID: W2345883768
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DOI: 10.1007/S00229-016-0844-0
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description: scientific article; zbMATH DE number 6666256