On the canonical degrees of curves in varieties of general type

TL;DR

This paper investigates bounds on the canonical degrees of curves in varieties of general type, providing explicit examples and proving inequalities in the context of Shimura varieties, thus advancing understanding of geometric constraints.

Contribution

It demonstrates that the constant in the degree bound must be at least the dimension of the ambient variety and proves the inequality for compact Shimura varieties.

Findings

Explicit examples from Shimura varieties show the constant must be at least the dimension.

Proves the canonical degree inequality for compact Shimura varieties.

Supports the conjecture relating degree bounds to geometric properties.

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.