A Hermite-Minkowski type theorem of varieties over finite fields

We show the finiteness of étale coverings of a variety over a finite field with given degree whose ramification bounded along an effective Cartier divisor. The proof is an application of P. Delgine's theorem (H. Esnault and M. Kerz, 2012) [4] on a finiteness of l-adic sheaves with restricted ramification. By applying our result to a smooth curve over a finite field, we obtain a function field analogue of the classical Hermite-Minkowski theorem.