Families of algebraic varieties and towers of curves over finite fields

A tower of algebraic curves is an infinite sequence of curves $C_n$ and finite morphisms $C_n\to C_{n-1}$, where we assume that the genus of $(C_n)$ is unbounded. I will speak about a new construction of towers of algebraic curves over finite fields. We begin with a family $X\to C$ of algebraic varieties over a projective curve $C$ that is smooth over an open subset $U$. The $i$-th derived étale direct image of the constant sheaf $Z/\ell^n Z$ corresponds to a local system on $U$. There is a fiberwise projectivisation of this local system, which is a generically étale scheme $U_n$ over $U$. We prove that under some strong technical conditions on the family $X$ the scheme $U_n$ is a geometrically irreducible algebraic curve over $U$. Let $C_n$ be the smooth compactification of $U_n$. Then the curves $C_n$ form a tower. I will give examples of interesting towers of this sort.