Diagonal property of the symmetric product of a smooth curve

Indranil Biswas; Sanjay Kumar Singh

doi:10.1016/j.crma.2015.02.007

Algebraic geometry

Diagonal property of the symmetric product of a smooth curve
[Propriété de la diagonale pour les produits symétriques d'une courbe lisse]

Présenté par : Claire Voisin

Indranil Biswas ¹ ; Sanjay Kumar Singh ²

¹ School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India
² Institute of Mathematics, Polish Academy of Sciences, Warsaw, 00656, Poland

Comptes Rendus. Mathématique, Volume 353 (2015) no. 5, pp. 445-448.

Résumé
Abstract

Soit C une courbe irréductible, lisse, définie sur un corps algébriquement clos. Nous montrons que le produit symétrique ${Sym}^{d} (C)$ a la propriété de la diagonale, pour tout $d \geq 1$ . Pour tous entiers n et r, soit $Q_{O_{C}^{\oplus n}} (n r)$ le schéma Quot paramétrant tous les quotients de torsion de $O_{C}^{\oplus n}$ de degré nr. Nous montrons que $Q_{O_{C}^{\oplus n}} (n r)$ a la propriété du point, faible.

Let C be an irreducible smooth projective curve defined over an algebraically closed field. We prove that the symmetric product ${Sym}^{d} (C)$ has the diagonal property for all $d \geq 1$ . For any positive integers n and r, let $Q_{O_{C}^{\oplus n}} (n r)$ be the Quot scheme parameterizing all the torsion quotients of $O_{C}^{\oplus n}$ of degree nr. We prove that $Q_{O_{C}^{\oplus n}} (n r)$ has the weak-point property.

Reçu le : 2015-01-14
Accepté le : 2015-02-25
Publié le : 2015-03-11

DOI : 10.1016/j.crma.2015.02.007

Affiliations des auteurs :

Indranil Biswas ¹ ; Sanjay Kumar Singh ²

¹ School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India
² Institute of Mathematics, Polish Academy of Sciences, Warsaw, 00656, Poland

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     author = {Indranil Biswas and Sanjay Kumar Singh},
     title = {Diagonal property of the symmetric product of a smooth curve},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {445--448},
     publisher = {Elsevier},
     volume = {353},
     number = {5},
     year = {2015},
     doi = {10.1016/j.crma.2015.02.007},
     language = {en},
}

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Indranil Biswas; Sanjay Kumar Singh. Diagonal property of the symmetric product of a smooth curve. Comptes Rendus. Mathématique, Volume 353 (2015) no. 5, pp. 445-448. doi : 10.1016/j.crma.2015.02.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.02.007/

Bibliographie
Cité par

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[3] E. Bifet; F. Ghione; M. Letizia On the Abel–Jacobi map for divisors of higher rank on a curve, Math. Ann., Volume 299 (1994), pp. 641-672

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[5] O. Debarre The diagonal property for abelian varieties, Curves and Abelian Varieties, Contemporary Mathematics, vol. 465, American Mathematical Society, Providence, RI, USA, 2008, pp. 45-50

[6] A. Grothendieck Techniques de construction et théorèmes d'existence en géométrie algébrique, IV. Les schémas de Hilbert. IV, Séminaire Bourbaki, vol. 6, Société mathématique de France, Paris, 1995, pp. 249-276 (Exp. No. 221)

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[8] P. Pragacz; V. Srinivas; V. Pati Diagonal subschemes and vector bundles, Pure Appl. Math. Q., Volume 4 (2008), pp. 1233-1278

Cité par Sources :

^☆ The first named author is supported by the J.C. Bose Fellowship. The second named author is supported by IMPAN Postdoctoral Research Fellowship.

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