## 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 $C$.