## B-N-R Part 4: Prym Varieties

The last post was on the generalities of Abelian varieties, and constructing a map.  This time, we’re going to do it for a specific one, and the maps involved will all be useful later.  We start out with $\pi:Y\to X$ a finite morphism of curves.

There’s a very powerful theorem which lets us do many things, called the Relative Duality Theorem.  I won’t be proving it, just stating it, and even if the statement is a bit beyond anything else we do, we only really need one consequence of it:

Theorem (Relative Duality) Let $f:Y\to X$ be a smooth morphism of smooth projective varieties of relative dimension $n$.  Let $\mathscr{F}$ be a coherent sheaf on $Y$ flat over $X$ and $\mathscr{G}$ be a coherent sheaf on $X$.  Suppose that for some $i\leq n$, $R^{n-i}f_*$ commutes with base change for $\mathscr{F}$.  Then there is a canonical isomorphism $\mathscr{E}xt^i_f(\mathscr{F},\omega_f\otimes f^*\mathscr{G})\cong \mathscr{H}om_{\mathscr{O}_X}(R^{n-i}f_*\mathscr{F},\mathscr{G})$, where $\omega_f=\omega_Y\otimes f^*\omega_X^{-1}$ and $\mathscr{E}xt_f^i(-,\mathscr{H})$ is the $i$th derived functor of the composition $f_*\circ \mathscr{H}om_{\mathscr{O}_Y}(-,\mathscr{H})$.

Now, if you unravel this for curves, as above, you’ll realize that $n=0$ so $i=0$, so all the derived functors drop out of the formula.  Fiddling with it an using the projection formula, you can derive that $\pi_*(\mathscr{O}_Y)^*$ is isomorphic to $\pi_*\omega_f$.  This is really the fact we want.  That and the fact that the ramification divisor $\Delta$ is a section of $\omega_f$.  We set $\mathfrak{D}=\det(\pi_*\mathscr{O}_Y)^{-1}$, which is thus also $\det\pi_*\omega_\pi$ and $\det\pi_*\mathscr{O}(\Delta)$.

Now, we shift gears for a bit and define a map $Nm:\mathrm{Pic}(Y)\to \mathrm{Pic}(X)$.  This map only makes sense for curves.  Take $L\in \mathrm{Pic}(Y)$.  It is isomorphic to $\mathscr{O}(D)$ for some divisor $D=\sum n_ip_i$ on $Y$.  We then look at $\pi(D)=\sum n_i\pi(p_i)$, which is a divisor on $X$.  We define $Nm(L)=\mathscr{O}(\pi(D))$.  I’ll leave it as an exercise to check that this is well defined.  But here’s the fun fact: $\det(\pi_*L)\cong Nm(L)\otimes\mathfrak{D}^{-1}$ for all $L\in \mathrm{Pic}(Y)$!  This implies that $Nm(\mathscr{O}(\Delta))=\mathfrak{D}^2$, also by simple algebraic manipulation.

This will all be useful later on, but now, we must define the Prym variety associated to a ramified covering of curves.  We have our morphism $\pi:Y\to X$, and we further assume that $\pi^*:J_X\to J_Y$ is injective.  Our ideal situation is that $J_Y=A$ and $\pi^*J_X=N$, so that teh orthogonal to $N$, which we’ll call $P$, is the kernel of map $Nm$, and that should be the Prym variety.  However, this required that we make quite a lot of choices, and we want things to be as canonical as possible.

So instead, we set $g=g_X$ and $h=g_Y$, and take $A'=J_Y^{h-1}$ and $N'=\pi^*J_X^{g-1}$, so that we have the Riemann Theta Divisors to work with.  We also choose $P'=Nm^{-1}(\mathfrak{D})$ with $Nm:J_Y^{\deg\mathfrak{D}}\to J_X^{\mathfrak{D}}$.  We get a map $N'\times P'\to A'$ by $(n,p)\mapsto n\otimes p$, and we can apply the stuff from the last post.  When the smoke clears, we find out that if we pull back $\Theta_Y$ along this map, its restriction to $N'\times\{\alpha\}$ turns out to be $\mathscr{O}(n\Theta_X)$, and we’ll denote by $\tau$ the restriction to $P'$.  Then the pullback of the canonical section of $\Theta_Y$ gives an isomorphism $\Gamma(P',\tau)^*\to \Gamma(J_X^{g-1},n\Theta_X)$, and we get the commuting diagram described last time, for the Prym variety of the cover.