Let $X$ be a smooth projective curve. How do I construct a coherent sheaf $\mathcal{F}$ on $\text{Pic}^n X$ (i.e., the component of the Picard scheme of $X$ parametrizing line bundles of degree $n$) such that$$\mathbb{P}(\mathcal{F}) := \text{Proj}\,\text{Sym}(\mathcal{F})$$equals the $n$th symmetric power $\text{Sym}^nX$?

(Here, "equals" means ``canonically isomorphic as a scheme over $\text{Pic}^nX$"; notice that $\text{Sym}^nX$ is equipped with a canonical map to $\text{Pic}^nX$ because $X$ is a smooth curve.)

Ideally, the construction of $\mathcal{F}$ should be as canonical as possible.