Let $C$ be a Riemann surface of genus $g$, $Sym^{n}C$ be the nth symmetric power of $C$ with $n\geq 2g$, and $JC$ denote the Jacobian of $C$.

Question: Is it generally true that $Sym^{n}C\cong JC\times\mathbb{P}^{n-g}$, where $\mathbb{P}^{n-g}$ is a projective bundle?

Thanks a lot