## Duke Mathematical Journal

### Monodromy of codimension 1 subfamilies of universal curves

Richard Hain

#### Abstract

Suppose that $g\ge3$, that $n\ge0$, and that $\ell\ge1$. The main result is that if $E$ is a smooth variety that dominates a codimension $1$ subvariety $D$ of $\mathcal{M}_{g,n}[\ell]$, 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 $\pi_{1}(E,e_{o})\to {\mathrm{Sp}}_{g}(\widehat{ \mathbb{Z}})$ has finite index in ${\mathrm{Sp}}_{g}(\widehat{ \mathbb{Z}})$. A similar result is proved for codimension $1$ families of principally polarized abelian varieties.

#### Article information

Source
Duke Math. J., Volume 161, Number 7 (2012), 1351-1378.

Dates
First available in Project Euclid: 4 May 2012

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1336142078

Digital Object Identifier
doi:10.1215/00127094-1593299

Mathematical Reviews number (MathSciNet)
MR2922377

Zentralblatt MATH identifier
1260.14014

#### Citation

Hain, Richard. Monodromy of codimension 1 subfamilies of universal curves. Duke Math. J. 161 (2012), no. 7, 1351--1378. doi:10.1215/00127094-1593299. https://projecteuclid.org/euclid.dmj/1336142078

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