On the canonical degrees of curves in varieties of general type

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.

10 pages, to appear in Geometric and Functional Analysis

References in corpus (2)

• A linear lower bound on the gonality of modular curves
• A more general abc conjecture

Cited by in corpus (4)

• Negative curves on algebraic surfaces
• A remark on the intersection of plane curves
• The exceptional set in Vojta's conjecture for algebraic points of bounded degree
• On finiteness of curves with high canonical degree on a surface