Diagonal property of the symmetric product of a smooth curve

Biswas, Indranil; Singh, Sanjay Kumar

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

Biswas, Indranil ¹ ; Singh, Sanjay Kumar 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, Tome 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 :

Biswas, Indranil ¹ ; Singh, Sanjay Kumar 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

@article{CRMATH_2015__353_5_445_0,
     author = {Biswas, Indranil and Singh, Sanjay Kumar},
     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},
     url = {http://archive.numdam.org/articles/10.1016/j.crma.2015.02.007/}
}

TY  - JOUR
AU  - Biswas, Indranil
AU  - Singh, Sanjay Kumar
TI  - Diagonal property of the symmetric product of a smooth curve
JO  - Comptes Rendus. Mathématique
PY  - 2015
SP  - 445
EP  - 448
VL  - 353
IS  - 5
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/j.crma.2015.02.007/
DO  - 10.1016/j.crma.2015.02.007
LA  - en
ID  - CRMATH_2015__353_5_445_0
ER  -

%0 Journal Article
%A Biswas, Indranil
%A Singh, Sanjay Kumar
%T Diagonal property of the symmetric product of a smooth curve
%J Comptes Rendus. Mathématique
%D 2015
%P 445-448
%V 353
%N 5
%I Elsevier
%U http://archive.numdam.org/articles/10.1016/j.crma.2015.02.007/
%R 10.1016/j.crma.2015.02.007
%G en
%F CRMATH_2015__353_5_445_0

Biswas, Indranil; Singh, Sanjay Kumar. Diagonal property of the symmetric product of a smooth curve. Comptes Rendus. Mathématique, Tome 353 (2015) no. 5, pp. 445-448. doi : 10.1016/j.crma.2015.02.007. http://archive.numdam.org/articles/10.1016/j.crma.2015.02.007/

Bibliographie
Cité par

[1] Baptista, J.M. On the $L^{2}$ -metric of vortex moduli spaces, Nucl. Phys. B, Volume 844 (2011), pp. 308-333

[2] Bertram, A.; Daskalopoulos, G.; Wentworth, R. Gromov invariants for holomorphic maps from Riemann surfaces to Grassmannians, J. Amer. Math. Soc., Volume 9 (1996), pp. 529-571

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

[4] Biswas, I.; Romão, N.M. Moduli of vortices and Grassmann manifolds, Commun. Math. Phys., Volume 320 (2013), pp. 1-20

[5] Debarre, O. 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] Grothendieck, A. 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)

[7] Nitsure, N. Construction of Hilbert and Quot schemes, Fundamental Algebraic Geometry, Mathematical Surveys and Monographs, vol. 123, American Mathematical Society, Providence, RI, USA, 2005, pp. 105-137

[8] Pragacz, P.; Srinivas, V.; Pati, V. 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.