Let $f:X\longrightarrow Y$ be a finite separable morphism of smooth projective integral curves over an algebraically closed field.


Then we have a linear equivalence of Weil divisors on $X$: $$ K\_X=f^\ast K\_Y + R.$$ Here $$R=\sum \textrm{length} (\Omega\_{X/Y})\_p [p]$$ is the ramification divisor on $X$. This is the Riemann-Hurwitz theorem.


We have a short exact sequence $$ 0 \longrightarrow \textrm{Pic}^0(X) \longrightarrow \textrm{Pic}(X) \longrightarrow \mathbf{Z} \longrightarrow 0,$$ where $\textrm{Pic}(X)\longrightarrow \mathbf{Z}$ is the degree map. 


We know what the Riemann-Hurwitz theorem tells us on the degree part of $\textrm{Pic}(X)$. It gives us the topological data $g(X)$ in terms of the degree of $R$, the genus of $Y$ and the degree of $f$.


But what does it tell us on $\textrm{Pic}^0(X)$?



What does the Riemann-Hurwitz formula tell us on the Picard variety
The answer is quite classical when $f \colon X \to Y$ is an unramified double cover.
In this case Riemann - Hurwitz formula gives 


$g(X)-1 = 2g(Y)-2$.


Consider the following three natural maps:


$f^\* \colon J(Y) \to J(X)$,


$Nm \colon \textrm{Pic}^0(X) \to \textrm{Pic}^0(Y), \quad Nm(\sum a\_ip\_i):= \sum a\_if(p\_i)$


$\tau \colon J(X) \to J(X)$,


where $f^\*$ is induced by the pull-back of $0$-cycles, $Nm$ is the norm map and $\tau$ is the involution induced by the double cover $f$.


Then 


1. $\textrm{Ker} \; f^\*=\langle L \rangle$, where $L$ is a point of order $2$ in $J(Y)$;
2. the connected component of $Nm^{-1}(0)$ containing the identity coincides with the image of $I-\tau$. It is an Abelian subvariety of $\textrm{Pic}^0(X)$ of dimension $g(Y)-1$, that is denoted by $\textrm{Prym}(X, \tau)$.


Moreover, under the identification of $\textrm{Pic}^0(X)$ with $J(X)$, the principal polarization of $J(X)$ restricts to twice a principal polarization on $\textrm{Prym}(X, \tau)$.


The geometry of Prym varieties is very rich. In particular, Riemann-Hurwitz identity


$K\_X =f^\*K\_Y$


induces subtle relations between the Theta divisor $\Theta$ of $X$ and the Theta divisor $\widetilde{\Theta}$ of $\textrm{Prym}(X, \tau)$. 


You can look at [Arbarello-Cornalba-Griffiths-Harris, Geometry of algebraic curves, Appendix C] or at [Birkenhake-Lange, Complex Abelian Varieties, Chapter 12] for further details.


In the general case, it is possible to define the so-called *generalized Prym varieties*, at least where $f \colon X \to Y$ is a tame Galois branched cover. Look for instance at the paper of MERINDOL


"Varietés de Prym d'un revetement galoisien [Prym varieties of a Galois covering]"


Journal Reine Angew. Math. 461 (1995), 49-61.


