Suppose that , that , and that . The main result is that if is a smooth variety that dominates a codimension subvariety of , the moduli space of -pointed, genus , smooth, projective curves with a level structure, then the closure of the image of the monodromy representation has finite index in . A similar result is proved for codimension families of principally polarized abelian varieties.
"Monodromy of codimension 1 subfamilies of universal curves." Duke Math. J. 161 (7) 1351 - 1378, 15 May 2012. https://doi.org/10.1215/00127094-1593299