Abel-Jacobi map and curvature of the pulled back metric

Let $X$ be a compact connected Riemann surface of genus at least two. The Abel-Jacobi map $\phi :{\mathrm{S}\mathrm{y}\mathrm{m}}^d(X)\to {\mathrm{P}\mathrm{i}\mathrm{c}}^d(X)$ is an embedding if $d$ is less than the gonality of $X$. We investigate the curvature of the pull-back, by $\phi$, of the flat metric on ${\mathrm{P}\mathrm{i}\mathrm{c}}^d(X)$. In particular, we show that when $d=1$, the curvature is strictly negative everywhere if $X$ is not hyperelliptic, and when $X$ is hyperelliptic, the curvature is nonpositive with vanishing exactly on the points of $X$ fixed by the hyperelliptic involution.