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 ith 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.

About Charles Siegel

Charles Siegel is currently a postdoc at Kavli IPMU in Japan. He works on the geometry of the moduli space of curves.
This entry was posted in Abelian Varieties, Big Theorems, Curves, Vector Bundles. Bookmark the permalink.

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