On the monodromy group of the family of smooth plane curves

Abstract

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.