Degree of a smooth curve in an abelian variety

Question

Let $A$ be an abelian variety, $g$ be a positive integer and $\mathcal{L}$ be an ample line bundle on $A$.

Question : Is there a real $r>0$ such that, for all smooth curve $C$ of genus $g$ in $A$, we have $\deg_\mathcal{L} C\leq r$ ?

Thanks a lot.

Answer 1

Well, this is surely false for elliptic curves, i.e. when $g=1$.

What is true is that, for any $t \in \mathbb{N}$, the number of elliptic curves in $A$ with $\deg _{\mathcal{L}} C \leq t$ is finite. However, it may happen that there exist elliptic curves on $A$ whose $\mathcal{L}$-degree is arbitrarily large.

This already happens in the case $A=E \times E$, where $E$ is an elliptic curve. Setting $$N_{A}(t):= \{\textrm{number of elliptic curves on A such that} \deg _{\mathcal{L}} C \leq t \},$$ some asymptotic upper bounds for $N_A(t)$ are available. In fact, computing $N_A(t)$ is the same as finding the number of solutions of certain diophantine equations.

For a discussion on this problem and bibliographical references you can look at the preprint by L. Guerra Elliptic curves of bounded degree in a polarized Abelian variety, http://arxiv.org/abs/1306.2007.