What does the Riemann-Hurwitz formula tell us on the Picard variety

Question

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 $\textrm{Pic}^0(X)$?

Answer 1

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.