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arXiv · 1003.3804

On the canonical degrees of curves in varieties of general type

Pascal Autissier·Antoine Chambert-Loir·Carlo Gasbarri
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Abstract

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 coming from the theory of Shimura varieties that in general, the constant $a$ has to be at least equal to the dimension of the ambient variety. We also prove the desired inequality in the case of compact Shimura varieties.

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Pascal Autissier, Antoine Chambert-Loir, Carlo Gasbarri. 2010-03-19. On the canonical degrees of curves in varieties of general type. https://doi.org/10.1007/s00039-012-0188-1

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