On the stability of the universal quotient bundle restricted to congruences of low degree of 𝔾(1,3)
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 9 (2010) no. 3, pp. 503-522.

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 .

Classification : 14J60, 14M07, 14M15
Arrondo, Enrique 1 ; Cobo, Sofía 1

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

[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