Monodromy of codimension 1 subfamilies of universal curves

Abstract

Suppose that $g\ge 3$, that $n\ge 0$, and that $\ell \ge 1$. The main result is that if $E$ is a smooth variety that dominates a codimension $1$ subvariety $D$ of ${\mathcal{M}}_{g,n}\left[\ell \right]$, the moduli space of $n$-pointed, genus $g$, smooth, projective curves with a level $\ell$ structure, then the closure of the image of the monodromy representation ${\pi }_1(E,e_o)\to {\mathrm{Sp}}_g\left(\mathrm{Ẑ}\right)$ has finite index in ${\mathrm{Sp}}_g\left(\mathrm{Ẑ}\right)$. A similar result is proved for codimension $1$ families of principally polarized abelian varieties.