This paper addresses the conjecture that the canonical degree degC(KX) of a curve C in a variety X of general type is bounded from above by an expression of the form A(2gC−2)+B, where gC is the geometric genus of C and A,B are some constants, with possible exceptions corresponding to curves lying in a strictly closed subset. In particular, an effective geometric upper bound of the constants A and B in the conjecture is of interest to the authors of this paper. A theorem of Y. Miyaoka [Publ. Res. Inst. Math. Sci. 44 (2008), no. 2, 403–417; MR2426352 (2009g:14043)] proves this conjecture for smooth curves in minimal surfaces with A>3/2. A conjecture of P. Vojta [Diophantine approximations and value distribution theory, Lecture Notes in Math., 1239, Springer, Berlin, 1987; MR0883451 (91k:11049)] claims that any constant A>1 is possible provided one restricts to curves of bounded gonality. This paper shows by explicit examples coming from the theory of Shimura varieties that in general, A should be at least dimX. This paper also proves the desired inequality in the case of compact Shimura varieties which are associated to a semisimple, connected and simply connected Q-anisotropic algebraic group over Q and which contain a totally geodesic compact Riemann surface. The constructions given in the paper use the images of a totally geodesic Riemann surface under Hecke correspondences and powers of Shimura curves.

Autissier, P., Chambert Loir, A., Gasbarri, C. (2012). On the canonical degrees of curves in varieties of general type. GEOMETRIC AND FUNCTIONAL ANALYSIS, 22(5), 1051-1061 [10.1007/s00039-012-0188-1].

On the canonical degrees of curves in varieties of general type

GASBARRI, CARLO
2012-01-01

Abstract

This paper addresses the conjecture that the canonical degree degC(KX) of a curve C in a variety X of general type is bounded from above by an expression of the form A(2gC−2)+B, where gC is the geometric genus of C and A,B are some constants, with possible exceptions corresponding to curves lying in a strictly closed subset. In particular, an effective geometric upper bound of the constants A and B in the conjecture is of interest to the authors of this paper. A theorem of Y. Miyaoka [Publ. Res. Inst. Math. Sci. 44 (2008), no. 2, 403–417; MR2426352 (2009g:14043)] proves this conjecture for smooth curves in minimal surfaces with A>3/2. A conjecture of P. Vojta [Diophantine approximations and value distribution theory, Lecture Notes in Math., 1239, Springer, Berlin, 1987; MR0883451 (91k:11049)] claims that any constant A>1 is possible provided one restricts to curves of bounded gonality. This paper shows by explicit examples coming from the theory of Shimura varieties that in general, A should be at least dimX. This paper also proves the desired inequality in the case of compact Shimura varieties which are associated to a semisimple, connected and simply connected Q-anisotropic algebraic group over Q and which contain a totally geodesic compact Riemann surface. The constructions given in the paper use the images of a totally geodesic Riemann surface under Hecke correspondences and powers of Shimura curves.
2012
Pubblicato
Rilevanza internazionale
Articolo
Esperti anonimi
Settore MAT/03 - GEOMETRIA
English
Autissier, P., Chambert Loir, A., Gasbarri, C. (2012). On the canonical degrees of curves in varieties of general type. GEOMETRIC AND FUNCTIONAL ANALYSIS, 22(5), 1051-1061 [10.1007/s00039-012-0188-1].
Autissier, P; Chambert Loir, A; Gasbarri, C
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2108/121055
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