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.

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, AG From the Beginning, Algebraic Geometry, Curves, Hodge Theory, MaBloWriMo. Bookmark the permalink.

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