On the monodromy group of the family of smooth quintic plane curves

We consider the space P-d of smooth complex projective plane curves of degree d. There is the tautological family of plane curves defined over P-d, which has an associated monodromy representation rho(d ): pi 1 ( P-d ) -> Mod(Sigma g) into the mapping class group of the fiber. For d <= 4, classical algebraic geometry implies the surjectivity of rho(d ) . For d >= 5, the existence of a (d - 3)( rd) root of the canonical bundle implies that rho d cannot be surjective. The main result of this paper is that for d = 5, the image of rho( 5) is as large as possible, subject to this constraint. This requires combining the algebro-geometric work of L & ouml;nne with Johnson's theory of the Torelli subgroup of Mod(Sigma g).