Given a smooth projective geometrically connected curve $C$, a symmetric product of $C$ has the structure of a projective bundle over the Jacobian of $C$ (e.g. see Symmetric powers of a curve = projective bundle over Jacobian, and the relative version thereof).
Special cases:
Suppose that $C$ is a hyperelliptic curve defined by $y^2 = f(x)$ for some squarefree polynomial $f$.
Is there a simpler method of turning a symmetric power $C^{(n)}$ into a projective bundle (or some other kind of fiber bundle in the Zariski topology) in this case?
Are there any other "natural" varieties we can construct using $C$ which have the structure of a projective bundle (or some other kind of fiber bundle in the Zariski topology) for this special case? Do they carry over to curves of the form $y^d = f(x)$ ($f$ not a $d^{\text{th}}$ power)?
A nonexample: Consider the map map $C \longrightarrow \mathbb{P}^1$ defined by $(x, y) \mapsto x$. Even the map induced on the spaces remaining after removing branch points and their preimages can't be a fiber bundle in the Zariski topology since a "section" would give a map which is not a morphism as it involves $\sqrt{f(x)}$.
Possible generalizations:
- Are there any interesting examples where a symmetric product of a higher dimensional variety has the structure of a projective bundle in the Zariski topology?
