Comptes Rendus
no. 9-10
Algebraic Geometry
A smooth surface of tame representation type
[Une surface lisse de type de représentation modéré]

Présenté par : Claire Voisin

Daniele Faenzi1 ; Francesco Malaspina2
1 Université de Pau et des pays de lʼAdour, BP 576, 64012 Pau cedex, France
2 Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
Comptes Rendus. Mathématique, Volume 351 (2013) no. 9-10, pp. 371-374.

Nous montrons que le produit dʼune droite et dʼune conique lisse, plongé dans P5 par Segre, est une variété projective de type modéré, autrement dit quʼil nʼy a sur cette variété que des familles de dimension 1 au plus de fibrés indécomposables ACM. À notre connaissance, il sʼagit du premier exemple de variété lisse projective de type modéré, mise à part la courbe elliptique, qui est de ce type dʼaprès le travail fondamental dʼAtiyah (1957).

We show that the Segre product of a line and a smooth conic, naturally embedded in P5, is a smooth projective surface of tame representation type, namely all continuous families of indecomposable ACM bundles have dimension one. To our knowledge, this is the first example of smooth projective variety of this kind, besides the elliptic curve, which is of tame representation type according to Atiyah (1957).

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2013.05.004
Daniele Faenzi 1 ; Francesco Malaspina 2

1 Université de Pau et des pays de lʼAdour, BP 576, 64012 Pau cedex, France
2 Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy 
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Daniele Faenzi; Francesco Malaspina. A smooth surface of tame representation type. Comptes Rendus. Mathématique, Volume 351 (2013) no. 9-10, pp. 371-374. doi : 10.1016/j.crma.2013.05.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.05.004/

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[6] D. Eisenbud; J. Herzog The classification of homogeneous Cohen–Macaulay rings of finite representation type, Math. Ann., Volume 280 (1988) no. 2, pp. 347-352

[7] D. Faenzi, F. Malaspina, The CM representation type of homogeneous spaces, in preparation.

[8] J.W. Hoffman; H.H. Wang Castelnuovo–Mumford regularity in biprojective spaces, Adv. Geom., Volume 4 (2004) no. 4, pp. 513-536

[9] R.M. Miró-Roig The representation type of rational normal scrolls, Rend. Circ. Mat. Palermo, Volume 62 (2013) no. 1, pp. 153-164

[10] R.M. Miró-Roig; J. Pons-Llopis Representation type of rational ACM surfaces in P4, Algebr. Represent. Theory (2013) (in press) | DOI

Cité par Sources :

☆ D.F. was partially supported by ANR-09-JCJC-0097-0 INTERLOW and ANR GEOLMI, F.M. was partially supported by Research Network Program GDRE-GRIFGA.

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