A ₁ obstruction to having finite index monodromy and an unusual subgroup of infinite index in Mod (g)

Let X be an algebraic surface with L an ample line bundle on X. Let (X, L) be the geometric monodromy group associated to family of nonsingular curves in X that are zero loci of sections of L. We provide obstructions to (X, L) being finite index in the mapping class group. We also show that for any k 0, the image of monodromy is finite index in appropriate subgroups of the quotient of the mapping class group by the kth term of the Johnson filtration assuming that 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.