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Comptes Rendus Mathématique
Volume 350, n° 9-10
pages 513-518 (mai 2012)
Doi : 10.1016/j.crma.2012.05.005
Received : 16 February 2012 ;  accepted : 9 May 2012
Courbes algébriques ordinaires et tissus associés
Ordinary algebraic curves and associated webs
 

Laurent Gruson a , Youssef Hantout b , Daniel Lehmann c
a Département de mathématiques, université de Versailles, 45, avenue des Etats Unis, 78035 Versailles, France 
b Département de mathématiques, université de Lille 1, 59650 Villeneuve dʼAscq cedex, France 
c 4, rue Becagrun, 30980 Saint Dionisy, France 

Résumé

Soit Γ une courbe algébrique complexe de degré d , intègre et non-dégénérée, dans lʼespace projectif complexe    . Notons   la dimension   de lʼespace vectoriel des polynômes homogènes de degré h en n variables, et   lʼentier ⩾1 tel que  . Nous appellerons ordinaires les courbes Γ ayant la propriété suivante : lʼensemble des hypersurfaces algébriques dʼun hyperplan « générique » H de  , qui sont de degré h et contiennent la section hyperplane  , est vide si  , et est un espace projectif de dimension égale à   si  . Pour  , ces courbes sont aussi celles dont le tissu associé dans   est ordinaire, au sens de Cavalier et Lehmann (2012) [[1]]. Leur genre arithmétique est majoré par le nombre    , et cette borne est atteinte pour celles de ces courbes ordinaires qui sont arithmétiquement de Cohen–Macaulay. Pour  , et tout degré  , la famille des courbes ordinaires de degré d qui sont arithmétiquement de Cohen–Macaulay est non vide et constitue une composante irréductible du schéma de Hilbert  .

Par contraste, les courbes intersections complètes de   hypersurfaces algébriques ne sont jamais ordinaires si  .

The full text of this article is available in PDF format.
Abstract

Let Γ be a complex algebraic curve of degree d , non-degenerate, reduced, and irreducible, in the complex projective space    . Denoting by   the dimension of the vector space of homogeneous polynomials of degree h with respect to n variables, let   be the integer (⩾1) such that  . The curve Γ is said to be ordinary if it has the following property: the set of algebraic hypersurfaces of a “generic” hyperplane H of   which have degree h and contain the hyperplane section  , is empty if  , and is a projective space of dimension   if  . Equivalently when  , the associated web in   of such a curve is “ordinary” in the sense of Cavalier and Lehmann (2012) [[1]]. The arithmetic genus of an ordinary curve is upper-bounded by the number    , and this bound is reached for these ordinary curves which are arithmetically Cohen–Macaulay. For   and any  , the family of the ordinary curves of degree d which are arithmetically Cohen–Macaulay is non-empty, and is an irreducible component of the Hilbert scheme  .

By contrast, the complete intersection of   algebraic hypersurfaces is never ordinary if  .

The full text of this article is available in PDF format.


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