Subextremal curves
type: Item
instance of: scholarly article
title: Subextremal curves
zbMATH Open document ID: 0918.14014
DOI: 10.1007/BF02677855
author: Scott Nollet
published in: Manuscripta Mathematica
publication date: 1998-05-13T00:00:00Z
full work available at URL: https://eudml.org/doc/156340
review text: \textit{M. Martin-Deschamps} and \textit{D. Perrin} obtained sharp bounds for the Rao function \(h^1(I_C(n))\) of a locally Cohen-Macaulay purely 1-dimensional curve \(C\subset\mathbb{P}^3\) [C. R. Acad. Sci., Paris Sér. I, 317, No. 12, 1159-1162 (1993; Zbl 0796.14029)] and characterized the extremal curves (i.e. the non-arithmetically Cohen-Macaulay curves achieving equality for their bound) in terms of certain Koszul modules in Ann. Sci. Éc. Norm. Supér., IV. Sér. 29, No. 6, 757-785 (1996; Zbl 0892.14005). In Boll. Unione Mat. Ital. (7), A 9, No. 3, 593-607 (1995; Zbl 0866.14020), \textit{P. Ellia} recovered the bounds of Martin-Deschamps and Perrin and characterized extremal curves as the schematic union of a line and of a plane curve. Moreover he proved that a curve \(C\) of degree \(d\leq 5\) and genus \(g\), which is neither arithmetically Cohen-Macaulay, nor extremal, verifies the stronger bound: \(h^1(I_C(n))\leq 1+(d-3)(d-4)/2-g\). All previous papers assumed the characteristic 0 hypothesis. In this article, without any particular assumption on the characteristic of the ground field, Scott Nollet improves Ellia's results, by proving, for neither arithmetically Cohen-Macaulay, nor extremal curves \(C\), the bounds: \[ h^1\bigl( I_C(n)\bigr)\leq\begin{cases} 0\quad & \text{if }n<1+g-(d-3)(d-4)/2\\ (d-3)(d-4)/2-g+n \quad & \text{if }1+g- (d-3)(d-4)/2\leq n<1\\ 1+(d-3)(d-4)-g\quad & \text{if }1\leq n\leq d-3\\ 1+(d-2)(d-3)/2-g-n\quad & \text{if }d-3<n\leq(d-2)(d-3)/2-g\\ 0\quad & \text{if }n>(d-2)(d-3)/2-g\end{cases} \] (theorem 2.11), proves that the above bounds are sharp and classifies subextremal curves (i.e. neither arithmetically Cohen-Macaulay, nor extremal curves achieving their bound) as the curves obtained from extremal ones by elementary biliaison of height 1 on a quadric surface (theorem 2.14). -- A key-tool to get the previous results is the notion of spectrum of a curve (see section 1), studied by \textit{Enrico Schlesinger} in his Ph. D. thesis […
Mathematics Subject Classification ID: 14H50
Mathematics Subject Classification ID: 14F25
Mathematics Subject Classification ID: 14M05
Mathematics Subject Classification ID: 14M06
zbMATH DE Number: 1135241
zbMATH Keywords: subextremal curves
zbMATH Keywords: biliaison
zbMATH Keywords: spectrum of a curve
zbMATH Keywords: Rao function for curves
reviewed by: Alberto Dolcetti
MaRDI profile type: Publication
cites work: Q3743413
cites work: Q4882699
cites work: Genre des courbes de l'espace projectif
cites work: Q4143433
cites work: Q3997640
cites work: Q4289154
cites work: Le schéma de Hilbert des courbes gauches localement Cohen-Macaulay n'est (presque) jamais réduit
cites work: Geometric invariants for liaison of space curves
cites work: The hilbert schemes of degree three curves
cites work: Q3703998
OpenAlex ID: W3192399916
Recommended article: On submaximal plane curves.
Recommended article: Smooth curves specialize to extremal curves
Recommended article: Q4938314
Recommended article: Q4267923
Recommended article: Asymptotic curves of submanifolds
Recommended article: Q4947439
Recommended article: Curvature of Approximating Curve Subdivision Schemes
Recommended article: On the total curvature of curves in nonsymmetric Minkowski spaces
Recommended article: Extreme Curvature of Polynomials
description: scientific article; zbMATH DE number 1135241