Sur Une Formule de Castelnuovo Pour Les Espaces Multisécants

Le Barz, Patrick

Bollettino dell'Unione Matematica Italiana, Tome 10-A (2007), p. 381-387 / Harvested from Biblioteca Digitale Italiana di Matematica

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Résumé
Abstract

Nello spazio proiettivo $P^{2k-2}$ , consideriamo il numero $v_{k}$ dei sottospazi $(k-2)$ -dimensionali, $k$ -secanti una curva C, di grado $n$ e genere $g$ . Nel 1889, una formula per $v_{k}$ è stata data da Castelnuovo [2]; in [1] si trova una dimostrazione moderna. In questo lavoro, mi propongo di costruire la funzione generatrice della serie $\sum_{k\geq 0}v_{k}t^{k}\in Z[[t]]$ , senza usare i risultati di Castelnuovo.

Let $v_{k}$ be the number of $(k-2)$ -dimensional subspaces of $P^{2k-2}$ which are $k$ -secant to a curve $C$ (of degree $n$ and genus $g$ ). Castelnuovo (1889) gave a formula for $v_{k}$ (see [2]); one has a modern proof in the monograph [1]. Here we give explicitly the generating function of the series $\sum_{k\geq 0}v_{k}t^{k}\in Z[[t]]$ , without using Castelnuovo's results.

Publié le : 2007-06-01

MR 2339448 | Zbl 1139.14042

@article{BUMI_2007_8_10B_2_381_0,
     author = {Patrick Le Barz},
     title = {Sur Une Formule de Castelnuovo Pour Les Espaces Multis\'ecants},
     journal = {Bollettino dell'Unione Matematica Italiana},
     volume = {10-A},
     year = {2007},
     pages = {381-387},
     zbl = {1139.14042},
     mrnumber = {2339448},
     language = {fr},
     url = {http://dml.mathdoc.fr/item/BUMI_2007_8_10B_2_381_0}
}

Le Barz, Patrick. Sur Une Formule de Castelnuovo Pour Les Espaces Multisécants. Bollettino dell'Unione Matematica Italiana, Tome 10-A (2007) pp. 381-387. http://gdmltest.u-ga.fr/item/BUMI_2007_8_10B_2_381_0/

Bibliographie

[1] Arbarello, E. - Cornalba, M. - Griffiths, P. - Harris, J., Geometry of Algebraic Curves, vol. I, Springer-Verlag (1985). | MR 770932 | Zbl 0559.14017

[2] Castelnuovo, G., Una applicazione della Geometria Enumerativa alle curve algebriche, Rendiconti Palermo III (1889), 27-37. | MR 3622298

[3] Donaldson, S., Instantons in Yang-Mills theory, The interface of Mathematics and Particle Physics, Clarendon Press, Oxford (1990), 59-75. | MR 1103130

[4] Fulton, W. - YOUNG TABLEAUX, London Math. Soc. Student Texts 35, Cambridge University Press (1997). | MR 1464693

[5] Hartshorne, R., Algebraic Geometry, Graduate Texts in Mathematics, Springer-Verlag (1977). | MR 463157

[6] Le Barz, P., Formules pour les espaces multisécants aux courbes algébriques, C. R. Ac. Sc. Paris, Ser. I, 340 (2005), 743-746. | MR 2141062 | Zbl 1073.14069

[7] Le Barz, P., Formules pour les trisécantes des surfaces algébriques, L'Enseigne- ment Mathématique, 33 (1987), 1-66. | MR 896383

[8] Lehn, M., Chern classes of tautological sheaves on Hilbert schemes of points on surfaces, Inv. Math., 136 (1999), 157-207. | MR 1681097 | Zbl 0919.14001

[9] Semple, J., Roth, L., Introduction to Algebraic Geometry, Clarendon Press, Oxford (1949). | MR 34048 | Zbl 0041.27903

[10] Tikhomirov, A. - Troshina, T., Top Segre class of a standard vector bundle on the Hilbert scheme $Hilb^{4}S$ of a surface S, Algebraic geometry and its applications Yaroslavl', Aspects of Maths. vol. E25, Vieweg-Verlag (1994), 205-226. | MR 1282030 | Zbl 0819.14004

[11] Vassallo, V., Justification de la méthode fonctionnelle pour les courbes gauches, Acta Mat., 172 (1994), 257-297. | MR 1278112 | Zbl 0842.14039