A $$\pi _1$$ obstruction to having finite index monodromy and an unusual subgroup of infinite index in $$\text {Mod}(\Sigma _g)$$

Abstract Let X be an algebraic surface with $$\mathcal {L}$$ an ample line bundle on X . Let $$\Gamma (X, \mathcal {L})$$ be the geometric monodromy group associated to a 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.