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

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     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://www.numdam.org/item/ASNSP_2010_5_9_3_503_0/}
}

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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://www.numdam.org/item/ASNSP_2010_5_9_3_503_0/

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