Families of special Weierstrass points

Letterio Gatto; Parham Salehyan

doi:10.1016/j.crma.2009.09.018

Algebraic Geometry

Families of special Weierstrass points
[Familles de points de Weierstrass speciaux]

Présenté par : Jean-Pierre Demailly

Letterio Gatto ¹ ; Parham Salehyan ²

¹ Dipartimento di Matematica, Politecnico di Torino, C.so Duca degli Abruzzi 24, 10129 (TO), Italy
² Departamento de Matemática, UNESP, Rua Cristóvão Colombo, 2265, Jardim Nazareth 15054-000 São José do Rio Preto, SP, Brazil

Comptes Rendus. Mathématique, Volume 347 (2009) no. 21-22, pp. 1295-1298.

Résumé
Abstract

L'objectif principal de cette Note est de montrer que les lieux de points de Weierstrass speciaux dans une famille générale de courbes lisses $X \to S$ de genre $g ⩾ 2$ peuvent être étudiés simplement en tirant en arrière le calcul de Schubert qui vit naturellement dans une fibrée opportune de Grassmann. En utilisant cette idée nous obtenons des nouveaux résultats concernant la décomposition de la classe dans $A_{*} (X)$ du lieu des points de Weierstrass qui ont poids au moins 3 comme somme des classes de points de Weierstrass avec suites particulières de lacunes.

The purpose of this Note is to show that loci of (special) Weierstrass points on the fibers of a family $π : X \to S$ of smooth curves of genus $g ⩾ 2$ can be studied by simply pulling back the Schubert calculus naturally living on a suitable Grassmann bundle over $X$ . Using such an idea we prove new results regarding the decomposition in $A_{*} (X)$ of the class of the locus of Weierstrass points having weight at least 3 as the sum of classes of Weierstrass points having “bounded from below” gaps sequences.

Reçu le : 2009-05-06
Accepté le : 2009-09-20
Publié le : 2009-10-08

DOI : 10.1016/j.crma.2009.09.018

Affiliations des auteurs :

Letterio Gatto ¹ ; Parham Salehyan ²

¹ Dipartimento di Matematica, Politecnico di Torino, C.so Duca degli Abruzzi 24, 10129 (TO), Italy
² Departamento de Matemática, UNESP, Rua Cristóvão Colombo, 2265, Jardim Nazareth 15054-000 São José do Rio Preto, SP, Brazil

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     author = {Letterio Gatto and Parham Salehyan},
     title = {Families of special {Weierstrass} points},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1295--1298},
     publisher = {Elsevier},
     volume = {347},
     number = {21-22},
     year = {2009},
     doi = {10.1016/j.crma.2009.09.018},
     language = {en},
}

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Letterio Gatto; Parham Salehyan. Families of special Weierstrass points. Comptes Rendus. Mathématique, Volume 347 (2009) no. 21-22, pp. 1295-1298. doi : 10.1016/j.crma.2009.09.018. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.09.018/

Bibliographie
Cité par

[1] W. Fulton Intersection Theory, Springer-Verlag, 1984

[2] L. Gatto (Notes on) Intersection Theory on Moduli Space of Curves, Monografas de Matemática, Instituto Nacional de Matemática Pura e Aplicada, Rio de Janeiro, 2000

[3] L. Gatto Schubert calculus via Hasse–Schmidt derivations, Asian J. Math., Volume 9 (2005) no. 3, pp. 315-322

[4] L. Gatto, Schubert calculus: An algebraic introduction, 25° Colóquio Brasileiro de Matemática, Instituto Nacional de Matemática Pura e Aplicada, Rio de Janeiro, 2005

[5] L. Gatto, A. Nigro, A remark on Porteous' formula (2009), in preparation

[6] L. Gatto, P. Salehyan, Weierstrass points of the universal curve (2009), in preparation

[7] L. Gatto; T. Santiago Schubert calculus on Grassmann algebra, Canad. Math. Bull., Volume 52 (2009) no. 2, pp. 200-212

[8] L. Gatto; F. Ponza Derivatives of Wronskians with applications to families of special Weierstrass points, Trans. Amer. Math. Soc., Volume 351 (1999) no. 6, pp. 2233-2255

[9] D. Laksov; A. Thorup A determinantal formula for the exterior powers of the polynomial ring, Indiana Univ. Math. J., Volume 56 (2007) no. 2, pp. 825-845

[10] D. Laksov; A. Thorup Schubert calculus on Grassmannians and exterior products, Indiana Univ. Math. J., Volume 58 (2009) no. 1, pp. 283-300

[11] R.F. Lax Weierstrass points of the universal curve, Math. Ann., Volume 216 (1975), pp. 35-42

[12] R.F. Lax Gap sequences and moduli in genus 4, Math. Z., Volume 175 (1980), pp. 67-75

[13] D. Mumford Towards an enumerative geometry of moduli space of curves, Arithmetic and Geometry, vol. II, Progr. Math., vol. 36, Birkhäuser Boston, Boston, MA, 1983, pp. 271-328

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