The kernel of the monodromy of the universal family of degree d smooth plane curves

We consider the parameter space U-d of smooth plane curves of degree d. The universal smooth plane curve of degree d is a fiber bundle epsilon(d) -> U-d with fiber diffeomorphic to a surface Sigma(g). This bundle gives rise to a monodromy homomorphism rho(d) : pi(1)(U-d) -> Mod(Sigma(g)), where Mod(Sigma(g)) := pi(0)(Diff(+)(Sigma(g))) is the mapping class group of Sigma(g). The main result of this paper is that the kernel of rho(4) : pi(1)(U-4) -> Mod(Sigma(3)) is isomorphic to F-infinity xZ/3Z, where F-infinity is a free group of countably infinite rank. In the process of proving this theorem, we show that the complement Teich(Sigma(g))\H-g of the hyperelliptic locus H-g in Teichmuller space Teich(Sigma(g)) has the homotopy type of an infinite wedge of spheres. As a corollary, we obtain that the moduli space of plane quartic curves is aspherical. The proofs use results from the Weil-Petersson geometry of Teichmuller space together with results from algebraic geometry.