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

Abstract We consider the space $\mathcal{P}_d$ of smooth complex projective plane curves of degree $d$ . There is the tautological family of plane curves defined over $\mathcal{P}_d$ , which has an associated monodromy representation $\rho _d: \pi _1(\mathcal{P}_d) \to \textrm{Mod}(\Sigma _g)$ into the mapping class group of the fiber. For $d \le 4$ , classical algebraic geometry implies the surjectivity of $\rho _d$ . For $d \ge 5$ , 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 Lönne with Johnson’s theory of the Torelli subgroup of $\textrm{Mod}(\Sigma _g)$ .