A $\pi_1$ obstruction to having finite index monodromy and an unusual subgroup of infinite index in $\textrm{Mod}(\Sigma_g)$

Abstract

Let $X$ be an algebraic surface with $\mathcal{L}$ an ample line bundle on $X$. Let $\Gamma(X, \mathcal{L})$ be the \emph{geometric monodromy} group associated to family of nonsingular curves in $X$ that are zero loci of sections of $\mathcal{L}$. We provide obstructions to $\Gamma(X, \mathcal{L})$ being finite index in the mapping class group. We also show that for any $k \ge 0$, the image of monodromy is finite index in appropriate subgroups of the quotient of the mapping class group by the $k$th term of the Johnson filtration assuming that $\mathcal{L}$ is sufficiently ample. This enables us to construct several subgroups of the mapping class group with unusual properties, in some cases providing the first examples of subgroups with those properties.