  We consider the parameter space $\mathcal U_d$ of smooth plane curves of
degree $d$. The universal smooth plane curve of degree $d$ is a fiber bundle
$\mathcal E_d\to\mathcal U_d$ with fiber diffeomorphic to a surface $\Sigma_g$.
This bundle gives rise to a monodromy homomorphism $\rho_d:\pi_1(\mathcal
U_d)\to\mathrm{Mod}(\Sigma_g)$, where
$\mathrm{Mod}(\Sigma_g):=\pi_0(\mathrm{Diff}^+(\Sigma_g))$ is the mapping class
group of $\Sigma_g$. The main result of this paper is that the kernel of
$\rho_4:\pi_1(\mathcal U_4)\to\mathrm{Mod}(\Sigma_3)$ is isomorphic to
$F_\infty\times\mathbb{Z}/3\mathbb{Z}$, where $F_\infty$ is a free group of
countably infinite rank. In the process of proving this theorem, we show that
the complement $\mathrm{Teich}(\Sigma_g)\setminus\mathcal{H}_g$ of the
hyperelliptic locus $\mathcal{H}_g$ in Teichm\"uller space
$\mathrm{Teich}(\Sigma_g)$ 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 Teichm\"uller space together with results from algebraic geometry.
