  We consider the space of smooth complex projective plane curves of degree d.
Defined over this is the tautological family of plane curves, and hence there
is a monodromy representation into the mapping class group of the fiber. We
show two results concerning this monodromy group. First, we show that the
presence of an invariant known as a "n-spin structure" constrains the image in
ways not predicted by previous work of Beauville. Second, we show that for
degree d=5, our invariant is the only obstruction for a mapping class to be
contained in the image. This requires combining the algebro-geometric work of
L\"onne with Johnson's theory of the Torelli subgroup of the mapping class
group.
