Jacobians of Curves

As promised in the last post, I’m making another go at MaBloWriMo…maybe others will as well.  I don’t know if I’m going to have a coherent topic over the course of this month, but I’ll be starting with abelian varieties associated to curves, so expect me to talk about generalizations of Prym varieties eventually (unless I get distracted by something else along the way).  Today, the basic case: Jacobians!

We will be working exclusively over $\mathbb{C}$.  Let $C$ be a curve.  Then we have a short exact sequence of sheaves $0\to \mathbb{Z}\to \mathcal{O}_C\stackrel{\exp}{\to}\mathcal{O}_C^\times\to 0$.  This gives us a long exact sequence on cohomology, which simplifies to $0\to \mathbb{Z}\to \mathbb{C}\to\mathbb{C}^\times\to H^1(C,\mathbb{Z})\to H^1(C,\mathcal{O}_C)\to \mathrm{Pic}(C)\to \mathbb{Z}\to 0$.  The last four terms break off, and if we set $J(C)=H^1(C,\mathcal{O}_C)/H^1(C,\mathbb{Z})$, we have $0\to J(C)\to \mathrm{Pic}(C)\to \mathbb{Z}\to 0$.

A fairly straightforward computation tells us that $H^1(C,\mathbb{Z})$ is a lattice in $H^1(C,\mathcal{O}_C)$, and so $J(C)$ is a complex torus.  On a complex torus, we have that $H^i(X,\mathbb{Z})\cong \bigwedge^i H^1(X,\mathbb{Z})$, and on $C$, we have a pairing $H^1(C,\mathbb{Z})\times H^1(C,\mathbb{Z})\to \mathbb{Z}$ given by cup product.  This is skew-symmetric, so gives $\lambda\in H^2(J(C),\mathbb{Z})$.

And just as for the curve, we have a sequence $0\to H^1(J,\mathbb{Z})\to H^1(J,\mathcal{O}_J)\to \mathrm{Pic}(J)\stackrel{c_1}{\to} H^2(J,\mathbb{Z})\to\ldots$ and it can be shown that $\lambda=c_1(\mathcal{O}(\Theta))$ for some divisor $\Theta$.  This isn’t uniquely defined, of course, but only up to $H^1(\mathcal{O})/H^1(\mathbb{Z})$, and as Jacobians and the curves they’re defined from have the same first cohomology groups, $\Theta$ is defined up to a point of $J(C)$! Also, any such divisor is ample, and $3\Theta$ defines an embedding of $J(C)$ into $\mathbb{P}^{3^g-1}$ of degree $3^g g!$.  So $J(C)$ is an abelian variety, that is, a projective group variety.

As $J(C)$ acts transitively on $\mathrm{Pic}^d(C)$, and they are all isomorphic as abstract varieties (that is, without worrying about a natural group structure), $\mathrm{Pic}^d(C)$ is a torsor over $J(C)$.  We can identify them all by picking a basepoint $p\in C$, and for any $D$ a divisor of degree $d$, use the map $\mathrm{Pic}^d(C)\to J(C)$ by $D\mapsto D-dp$.

Tomorrow, we’ll talk about special loci in $\mathrm{Pic}^d(C)$ for $0\leq d\leq g-1$. 