nLab theta divisor

Idea

A certain divisor $\Theta$ definable on any abelian variety. It is defined by the zero-locus of the Riemann theta function of the abelian variety.

On Jacobian varieties it induces a principally polarized variety structure. In this case, the theta divisor obtains an alternate description. Suppose $C$ is a curve, and $A = J(C)$ is its Jacobian. For a basepoint $p$ on $C$, there is a natural map $u_p:C \to J(C)$ given by sending a point $q$ to the linear equivalence class of $q - p$ consisting of divisors of degree $0$. Then, using the abelian variety structure on $J(C)$, we may add $C$ to itself $k$ times on $J(C)$, giving rise to codimension-$k$ varieties $W_k(C)$. Riemann proved that $W_{g-1}(C)$ can be identified with $\Theta$ up to translation.

References

• Wikipedia, Theta divisor