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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 
$\textrm{Mod}(\Sigma _g)$.
