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.
