Algebraic Geometry
A smooth surface of tame representation type
Comptes Rendus. Mathématique, Volume 351 (2013) no. 9-10, pp. 371-374.

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).

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).

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2013.05.004
Faenzi, Daniele 1; Malaspina, Francesco 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
@article{CRMATH_2013__351_9-10_371_0,
     author = {Faenzi, Daniele and Malaspina, Francesco},
     title = {A smooth surface of tame representation type},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {371--374},
     publisher = {Elsevier},
     volume = {351},
     number = {9-10},
     year = {2013},
     doi = {10.1016/j.crma.2013.05.004},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2013.05.004/}
}
TY  - JOUR
AU  - Faenzi, Daniele
AU  - Malaspina, Francesco
TI  - A smooth surface of tame representation type
JO  - Comptes Rendus. Mathématique
PY  - 2013
SP  - 371
EP  - 374
VL  - 351
IS  - 9-10
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2013.05.004/
DO  - 10.1016/j.crma.2013.05.004
LA  - en
ID  - CRMATH_2013__351_9-10_371_0
ER  - 
%0 Journal Article
%A Faenzi, Daniele
%A Malaspina, Francesco
%T A smooth surface of tame representation type
%J Comptes Rendus. Mathématique
%D 2013
%P 371-374
%V 351
%N 9-10
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2013.05.004/
%R 10.1016/j.crma.2013.05.004
%G en
%F CRMATH_2013__351_9-10_371_0
Faenzi, Daniele; Malaspina, Francesco. 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. http://www.numdam.org/articles/10.1016/j.crma.2013.05.004/

[1] Atiyah, M.F. Vector bundles over an elliptic curve, Proc. Lond. Math. Soc. (3), Volume 7 (1957) no. 7, pp. 414-452

[2] Ballico, E.; Malaspina, F. Qregularity and an extension of the Evans–Griffiths criterion to vector bundles on quadrics, J. Pure Appl. Algebra, Volume 213 (2009) no. 2, pp. 194-202

[3] Ballico, E.; Malaspina, F. Regularity and cohomological splitting conditions for vector bundles on multiprojectives spaces, J. Algebra, Volume 345 (2011), pp. 137-149

[4] Costa, L.; Miró-Roig, R.M.; Pons-Llopis, J. The representation type of Segre varieties, Adv. Math., Volume 230 (2012) no. 4–6, pp. 1995-2013

[5] Drozd, Y.; Greuel, G.-C. Tame and wild projective curves and classification of vector bundles, J. Algebra, Volume 246 (2011) no. 1, pp. 1-54

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

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

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

Cited by 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.