The Jacobian, Abel's theorem, and the Abel--Jacobi map

The Jacobian geometrizes degree-zero divisors: Abel kills exactly the principal ones, and Jacobi inversion reaches every divisor class.Suggest a slogan

For $X$ a smooth projective curve of genus $g$ over $\CC$, \[ \Jac(X) = \dualof{H^0(X,\Omega^1)} / H_1(X,\ZZ) , \] a complex torus of dimension $g$, and an abelian variety. The Abel–Jacobi map sends a degree-zero divisor $\sum (p_i - q_i)$ to $\sum \int_{q_i}^{p_i}$, a linear functional on $H^0(\Omega^1)$ taken modulo periods.

(Abel)

A divisor of degree zero is principal exactly when its image under the Abel–Jacobi map is zero. Hence $\Pic^0(X) \cong \Jac(X)$.

(Jacobi inversion)

The Abel–Jacobi map is surjective, so $\Jac(X)$ is exactly the group of degree-zero divisor classes.

The significance is the sentence the two theorems combine to: the Jacobian is a projective variety whose points are the degree-zero line bundles on $X$, so a discrete-looking classification problem becomes a geometric object of dimension $g$.

Fixing a base point $p_0$ gives $X \to \Jac(X)$, $p \mapsto [p - p_0]$, which is an embedding for $g \geq 1$. In genus $1$ it is an isomorphism: an elliptic curve is its own Jacobian, and the group law on $E$ is the statement that $[p]+[q]=[r]+[0]$ exactly when $p+q=r$. Everything special about elliptic curves is this coincidence of $X$ with $\Jac X$.

State Abel's theorem.

What is the significance of the Jacobian?

What kind of map is the Abel--Jacobi map, and what is it in genus $1$?

Dependencies

T-MWDVLRiemann--Roch and Serre duality on a curveuses
PR-Y5S7VThe degree of a divisor on a projective curveuses

Backlinks

T-CRVUNIFEvery elliptic curve over $\CC$ is a torus $\CC/\Lambda$related-to
T-CRVJACFUNThe Jacobian represents $\Pic^0(X/-)$, and is a smooth proper group scheme of dimension $g$related-to
PR-CRVGRPThe group law on an elliptic curve, and multiplication by $n$uses

What links to this

Embeddings and Jacobians