Subvarieties of Jacobians

For this whole post, we’ll take C to be a curve and J=J(C) the Jacobian of the curve.  We’re going to construct several special subvarieties (not special in any technical sense, though) of J, which encode a great deal of geometric information about C.

We start with a problem: is there a natural map C\to J? The answer is no, sadly.  However, all hope is not lost! Nay, there IS a natural map C\to \mathrm{Pic}^1(C), given by taking each point p\in C to the divisor represented by p.  For g>0, this map is an isomorphism onto its image (injectivity is easy, as two points being linearly equivalent implies genus zero, smoothness of the image is a bit less obvious, but not hard).  So we know that C sits inside of J, but that requires some identification of J with \mathrm{Pic}^1(C).

In fact, for all d\geq 0, we have a map C^d\to \mathrm{Pic}^d(C) given by taking (p_1,\ldots,p_d)\mapsto p_1+\ldots+p_d.  This even factors through \mathrm{Sym}^d(C)=C^d/S_d, because the addition on \mathrm{Pic}(C) is abelian.  We will call these the level d Abel-Jacobi maps, and if d=1, we call it the Abel-Jacobi map, and denote them by \mathcal{AJ}_d, and drop the subscript for d=1.  We call a curve in J an Abel-Jacobi curve if it is of the form \mathcal{AJ}(C)-D where D\in \mathrm{Pic}^1(C).

Moving to higher dimensions, we denote the image of \mathcal{AJ}_d by W_d, which is the locus of effective divisors in \mathrm{Pic}^d(C).  For any D\in W_d, the fiber is the set of all divisors D' linearly equivalent to it, and thus is a projective space, naturally identified with \mathbb{P}H^0(C,\mathcal{O}(D))=|D|.

We can define a function \phi(D)=h^0(C,\mathcal{O}(D))-1, which is just the dimension of the fiber over D.  This function is upper semicontinuous, that is, the loci where \phi(D)\geq r are all closed.  We define these loci to be W_d^r, that is, W_d^r is the locus of all complete linear systems of degree d and rank r on C.  We’ll return to these later.  For now, we’ll study W_{g-1}\subset \mathrm{Pic}^{g-1}(C).

Here, the codimension is 1, so we have a divisor, as \mathrm{Pic}^{g-1}(C) is a smooth variety.  The Riemann-Roch formula tells us that for D\in \mathrm{Pic}^{g-1}(C), we have h^0(D)=h^0(K-D), so D\mapsto K-D is an involution of W_{g-1}, which we’ll denote by \iota.  We’ll use the same letter for the involution applied to arbitrary points of \mathrm{Pic}^{g-1}(C).  Now, if we take a symmetric theta divisor, that is, \Theta as before, such that if D\in \Theta then -D\in \Theta, for D\in J, then we get a nice theorem:

Riemann’s Theorem: There exists a constant \kappa\in \mathrm{Pic}^{g-1}(C) such that \Theta+\kappa=W_{g-1}.

This theorem gives us a geometric model of the theta divisor, which we can now construct explicitly from the curve.  It’s the first step in a program that ties together the geometry of the pair (J,\Theta) with the geometry of C, and culminates in the Riemann Singularity Theorem and the Torelli Theorem.  We’ll pick up these threads later.

For now, we’re going to look more closely at \kappa.  If we denote translation by \kappa as T_\kappa, then we can write Riemann’s Theorem as T^*_\kappa \Theta=W_{g-1}.  Now, looking at \iota^*T_\kappa^*(-1)^*\Theta, we note that symmetry of \Theta, Riemann’s Theorem, and the fact that W_{g-1} is \iota-invariant, tells us that (-1)T_\kappa \iota=T_{\omega\otimes \kappa^{-1}}, and so \kappa\cong \omega_C\otimes \kappa^{-1}.  Thus, \kappa is a line bundle satisfying \kappa^2\cong \omega_C.  Such line bundles are called theta characteristics, and these turn out to be extremely closely tied to the geometry of the curve C.

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. Bookmark the permalink.

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