AUTISSIER, Pascal; CHAMBERT-LOIR, Antoine; GASBARRI, Carlo

doi:10.1007/s00039-012-0188-1

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AUTISSIER, Pascal
Institut de Mathématiques de Bordeaux [IMB]

CHAMBERT-LOIR, Antoine
Institut de Recherche Mathématique de Rennes [IRMAR]

GASBARRI, Carlo
Institut de Recherche Mathématique Avancée [IRMA]

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Geometric And Functional Analysis. 2012, vol. 22, n° 5, p. 1051-1061

Springer Verlag

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A widely believed conjecture predicts that curves of bounded geometric genus lying on a variety of general type form a bounded family. One may even ask whether the canonical degree of a curve $C$ in a variety of general type is bounded from above by some expression $a\chi(C)+b$, where $a$ and $b$ are positive constants, with the possible exceptions corresponding to curves lying in a strict closed subset (depending on $a$ and $b$). A theorem of Miyaoka proves this for smooth curves in minimal surfaces, with $a>3/2$. A conjecture of Vojta claims in essence that any constant $a>1$ is possible provided one restricts oneself to curves of bounded gonality. We show by explicit examples that in general, the constant $a$ has to be at least equal to the dimension of the ambient variety.< Réduire

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On the canonical degrees of curves in varieties of general type

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