On the stability of the universal quotient bundle restricted to congruences of low degree of $\mathbb{G}{\bf (1,3)}$

Arrondo, Enrique; Cobo, Sofía

On the stability of the universal quotient bundle restricted to congruences of low degree of

𝔾 (1, 3)

Arrondo, Enrique ¹ ; Cobo, Sofía ¹

¹ Departamento de Álgebra, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 9 (2010) no. 3, pp. 503-522.

Résumé

We study the semistability of ${Q |}_{S}$ , the universal quotient bundle on $𝔾$ (1,3) restricted to any smooth surface $S$ (called congruence). Specifically, we deduce geometric conditions for a congruence $S$ , depending on the slope of a saturated linear subsheaf of ${Q |}_{S}$ . Moreover, we check that the Dolgachev-Reider Conjecture (i.e. the semistability of ${Q |}_{S}$ for nondegenerate congruences $S$ ) is true for all the congruences of degree less than or equal to 10. Also, when the degree of a congruence $S$ is less than or equal to 9, we compute the highest slope reached by the linear subsheaves of ${Q |}_{S}$ .

MR Zbl

Classification : 14J60, 14M07, 14M15

Affiliations des auteurs :

Arrondo, Enrique ¹ ; Cobo, Sofía ¹

¹ Departamento de Álgebra, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain

@article{ASNSP_2010_5_9_3_503_0,
     author = {Arrondo, Enrique and Cobo, Sof{\'\i}a},
     title = {On the stability of the universal quotient bundle restricted to congruences of low degree of $\mathbb{G}{\bf (1,3)}$},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {503--522},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 9},
     number = {3},
     year = {2010},
     mrnumber = {2722653},
     zbl = {1202.14038},
     language = {en},
     url = {http://archive.numdam.org/item/ASNSP_2010_5_9_3_503_0/}
}

TY  - JOUR
AU  - Arrondo, Enrique
AU  - Cobo, Sofía
TI  - On the stability of the universal quotient bundle restricted to congruences of low degree of $\mathbb{G}{\bf (1,3)}$
JO  - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY  - 2010
SP  - 503
EP  - 522
VL  - 9
IS  - 3
PB  - Scuola Normale Superiore, Pisa
UR  - http://archive.numdam.org/item/ASNSP_2010_5_9_3_503_0/
LA  - en
ID  - ASNSP_2010_5_9_3_503_0
ER  -

%0 Journal Article
%A Arrondo, Enrique
%A Cobo, Sofía
%T On the stability of the universal quotient bundle restricted to congruences of low degree of $\mathbb{G}{\bf (1,3)}$
%J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
%D 2010
%P 503-522
%V 9
%N 3
%I Scuola Normale Superiore, Pisa
%U http://archive.numdam.org/item/ASNSP_2010_5_9_3_503_0/
%G en
%F ASNSP_2010_5_9_3_503_0

Arrondo, Enrique; Cobo, Sofía. On the stability of the universal quotient bundle restricted to congruences of low degree of $\mathbb{G}{\bf (1,3)}$. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 9 (2010) no. 3, pp. 503-522. http://archive.numdam.org/item/ASNSP_2010_5_9_3_503_0/

Bibliographie
Cité par

[1] E. Arrondo, M.Bertolini and C. Turrini, A focus on focal surfaces, Asian J. Math. 3 (2001), 535–560. | MR | Zbl

[2] E. Arrondo and M. Gross, On smooth surfaces in $G (1, 𝐏^{3})$ with a fundamental curve, Manuscripta Math. 79 (1993), 283–298. | EuDML | Zbl

[3] E. Arrondo and I. Sols, “On Congruences of Lines in the Projective Space”, Mém. Soc. Math. France, Vol. 50, 1992. | Numdam | MR | Zbl

[4] S. Cobo, Simplicity of the universal quotient bundle restricted to congruences of lines in $¶^{3}$ , Adv. Geom. 6 (2006), 467–473. | MR | Zbl

[5] S. Cobo, “Estabilidad del Fibrado Universal Restringido a Congruencias”, PhD Thesis, Universidad Complutense de Madrid, 2008.

[6] P. Deligne and N. Katz, “Groupes de Monodromie en Géométrie Algébrique”, SGA7II, Springer LNM 340, 1973. | MR

[7] I. Dolgachev and I. Reider, On rank 2 vector bundles with $c_{1}^{2} = 10$ and $c_{2} = 3$ on Enriques surfaces, In: “Algebraic Geometry” (Chicago, IL), Lecture notes in Mahtematics, Springer-Verlag, Vol. 1479, 1991. | Zbl

[8] M. Gross, The distribution of bidegrees of smooth surfaces in $𝔾 (1, 3)$ , Math. Ann. 292 (1992), 127–147. | EuDML | MR | Zbl

[9] M. Gross, Surfaces of degree $10$ in the Grassmannian of lines in $3$ -space, J. Reine Angew. Math. 436 (1993), 87–127. | EuDML | MR | Zbl

[10] R. Hartshorne, “Algebraic Geometry”, Springer, 1997. | MR | Zbl

[11] M. Nagata, On self-intersection number of a section on a ruled surface, Nagoya Math. J. 37 (1970), 191–196. | MR | Zbl