Prym Varieties

Let ${\pi:\tilde{C}\rightarrow C}$ be an unramified double cover, where ${C}$ is geneus ${g}$ . Then ${\tilde{C}}$ has genus ${2g-1}$ by the Riemann-Hurwitz formula. Now, ${J(C)}$ encodes lots of information about the geometry of ${C}$ , especially with the additional data of the theta divisor. It turns out that for double covers, there’s an abelian variety that contains a lot of this data.

We, in fact, can describe this abelian variety in many ways. The simplest is probably that we have a pullback map ${\pi^*:J(C)\rightarrow J(\tilde{C})}$ , and we actually get a short exact sequence of abelian varieties ${0\rightarrow J(C)\stackrel{\pi^*}{\rightarrow}J(\tilde{C})\rightarrow P\rightarrow 0}$ , and ${P}$ is called the Prym variety associated to the cover.

Another description is that we can define a map ${Nm:J(\tilde{C})\rightarrow J(C)}$ , given by ${Nm(\sum n_P P)=\sum n_P \pi(P)}$ . Then, we look at the fiber over zero. The kernel here actually breaks up into two components. The component containing ${0}$ is isomorphic to ${P}$ , and the other is just a translation of ${P}$ .

So the first real theorem about Prym varieties is the following:

Wirtinger’s Theorem: The theta divisor on $\tilde{C}$ induces twice a principal polarization on $P$ . That is, $\tilde{\Theta}|_P\cong 2\Xi$ .

So if we set $\mathcal{RM}_g$ the space of unramified double covers of curves of genus $g$ , and $\mathcal{A}_g$ the space of principally polarized abelian varieties of dimension $g$ , then we have a map $P:\mathcal{RM}_g\to \mathcal{A}_{g-1}$ .

This leads to one of my favorite theorems

Theorem: The closure of the image of $P$ contains the Jacobians of genus $g-1$ .

To see this, we’ll use what are called Wirtinger covers. If $C\in \mathcal{M}_{g-1}$ , then $X=C/p\sim q$ is a stable curve of genus $g$ . We can then set $\tilde{X}=C_1\coprod C_2/p_1\sim q_2,p_2\sim q_1$ . Then we’ll need to work out the Jacobians of $X$ and $\tilde{X}$ . For $\tilde{X}$ , a line bundle is the same as one on $C$ , along with a nonzero complex number, that is, a map identifying the fibers over $p$ and $q$ . So $0\to \mathbb{C}^\times\to J(X)\to J(C)\to 0$ . Similarly, we have $0\to \mathbb{C}^\times\to \mathbb{C}^\times\times\mathbb{C}^\times\to J(\tilde{X})\to J(C)\times J(C)\to 0$ , and we get vertical maps given by multiplication, norm and tensor product. This then presents $0\to G\to \ker \mathrm{Nm}\to J(C)\to 0$ where $G$ is a finite group, and so the connected component of the identity is $J(C)$ . So allowing the double cover to degenerate gives Jacobians!

Thus, the Prym map is closely tied to the problem of figuring out which abelian varieties are Jacobians (for more details, see my ICTP talk).

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.

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Prym Varieties

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