On the stability of the universal quotient bundle restricted to congruences of low degree of <span class="mathjax-formula">$\mathbb{G}{\bf (1,3)}$</span>

Arrondo, Enrique; Cobo, Sofía

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

𝔾 (1, 3)

Arrondo, Enrique ; Cobo, Sofía

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 2722653 | Zbl 1202.14038

Classification : 14J60, 14M07, 14M15

@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},
     zbl = {1202.14038},
     mrnumber = {2722653},
     language = {en},
     url = {http://www.numdam.org/item/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://www.numdam.org/item/ASNSP_2010_5_9_3_503_0/

Bibliographie

[1] E. Arrondo, M.Bertolini and C. Turrini, A focus on focal surfaces, Asian J. Math. 3 (2001), 535–560. | MR 1868579 | Zbl 1027.14024

[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 155842 | Zbl 0803.14019

[3] E. Arrondo and I. Sols, “On Congruences of Lines in the Projective Space”, Mém. Soc. Math. France, Vol. 50, 1992. | Numdam | MR 1180696 | Zbl 0804.14016

[4] S. Cobo, Simplicity of the universal quotient bundle restricted to congruences of lines in $¶^{3}$ , Adv. Geom. 6 (2006), 467–473. | MR 2248262 | Zbl 1112.14050

[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 354657

[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 0766.14031

[8] M. Gross, The distribution of bidegrees of smooth surfaces in $𝔾 (1, 3)$ , Math. Ann. 292 (1992), 127–147. | EuDML 164906 | MR 1141789 | Zbl 0741.14017

[9] M. Gross, Surfaces of degree $10$ in the Grassmannian of lines in $3$ -space, J. Reine Angew. Math. 436 (1993), 87–127. | EuDML 153498 | MR 1207282 | Zbl 0761.14012

[10] R. Hartshorne, “Algebraic Geometry”, Springer, 1997. | MR 463157 | Zbl 0532.14001

[11] M. Nagata, On self-intersection number of a section on a ruled surface, Nagoya Math. J. 37 (1970), 191–196. | MR 258829 | Zbl 0193.21603