Fay's trisecant identity

In algebraic geometry, Fay's trisecant identity is an identity between theta functions of Riemann surfaces introduced by. Fay's identity holds for theta functions of Jacobians of curves, but not for theta functions of general abelian varieties.

The name "trisecant identity" refers to the geometric interpretation given by, who used it to show that the Kummer variety of a genus g Riemann surface, given by the image of the map from the Jacobian to projective space of dimension 2g – 1 induced by theta functions of order 2, has a 4-dimensional space of trisecants.

Statement
Suppose that
 * C is a compact Riemann surface
 * g is the genus of C
 * θ is the Riemann theta function of C, a function from Cg to C
 * E is a prime form on C&times;C
 * u,v,x,y are points of C
 * z is an element of Cg
 * ω is a 1-form on C with values in Cg

The Fay's identity states that

$$ \begin{align} &E(x,v)E(u,y)\theta\left(z+\int_u^x\omega\right)\theta\left(z+\int_v^y\omega\right)\\ - &E(x,u)E(v,y)\theta\left(z+\int_v^x\omega\right)\theta\left(z+\int_u^y\omega\right)\\ = &E(x,y)E(u,v)\theta(z)\theta\left(z+\int_{u+v}^{x+y}\omega\right) \end{align} $$

with

$$ \begin{align} &\int_{u+v}^{x+y}\omega=\int_u^x\omega+\int_v^y\omega=\int_u^y\omega+\int_v^x\omega \end{align} $$