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

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Of course, you can compute the dimension of $Pic^0(X)$ from this, because it's just the genus $g(X)$. Perhaps you know this already, but I can't tell what kind of information you are after. –  Donu Arapura Oct 29 '10 at 21:55
    
Well I'm not really after anything here. But it'd be nice to see what kind of "geometric" information we get on the Picard variety. That is, in my opinion, it should say something more about the Picard variety than just give its dimension. More precisely, when you consider the linear equivalence above modulo the degree you get points of the Picard variety. What are these and what does the equality mean? –  Ariyan Javanpeykar Oct 30 '10 at 6:12

1 Answer 1

up vote 8 down vote accepted

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.

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