What is the Kodaira dimension of symmetric products of curves? That is, given a projective smooth, connected complex curve $C$, what is the Kodaira dimension of $C^{(d)}=C^d/\mathfrak S\_d$?


When $d> g$, the genus of $C$, then $C^{(d)}$ is a bundle in projective spaces over the Jacobian of $C$, hence all pluriforms on $C^{(d)}$ vanish on the fibers of this fibration and $\kappa(C^{(d)})=-\infty$ in this case.
(Said in another way, $C^{(d)}$ is uniruled.) 


*Is something known for $2\leq d\leq g-1$?* (This question is prompted by [this other post](https://mathoverflow.net/questions/123862/rational-points-on-surfaces-of-general-type).) I suspect that the answer will strongly depend on fine properties of the curve $X$ (gonality, Brill-Noether properties) and there might not be a general and neat answer.


Perhaps surprisingly, the analogous question in higher dimensions is quite different since if $X$ is a projective smooth connected variety of dimension $>1$, the Kodaira dimension of (any desingularization) of $X^{(d)}$ is equal to $d \kappa(X)$, where $\kappa(X)$ is the Kodaira dimension of $X$ (D. Arapura, S. Archava, *[Kodaira dimension of symmetric powers](http://www.ams.org/journals/proc/2003-131-05/S0002-9939-02-06797-7/S0002-9939-02-06797-7.pdf)*, Proc. AMS, 2003).


[Edited to correct an incorrect assertion, as pointed out by @pbelmans]



Kodaira dimension of symmetric products of curves
Let $C$ be a smooth projective connected complex curve of genus $\geq 2$. Let me show that $C^{(d)}$ is of general type if $1\leq d\leq g-1$.


Equivalently, one needs to show that the image $W\_d$ of $C^{(d)}$ in the jacobian $J(C)$ is of general type, because $C^{(d)}\to W\_d$ is birational. If $W\_d$ were not of general type, then, by [Ueno, Classification of algebraic varieties I, Theorem 3.10], there would be a non-trivial abelian variety $A\subset J(C)$ such that $W\_d$ is stable by translation by $A$ (this is the argument in MP's comment above). But then, $W\_{g-1}$ would also be stable by translation by $A$. Now choose a point $x$ outside of $W\_{g-1}$ and consider the orbit $A.x$ : it is a positive-dimensional variety avoiding $W\_{g-1}$. This is a contradiction because $W\_{g-1}$ is an ample divisor: the theta divisor.


