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 $2^g-1$ 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 $C$
*$\theta$ is the Riemann theta function of $C$, a function from $\mathbb^g$ to $\mathbb$
*$E$ is a prime form on $C\times C$
*$u$, $v$, $x$, $y$ are points of $C$
*$z$ is an element of $\mathbb^g$
*$\omega$ is a 1-form on $C$ with values in $\mathbb^g$

The Fay's identity states that

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

with

$\begin &\int_^\omega=\int_u^x\omega+\int_v^y\omega=\int_u^y\omega+\int_v^x\omega \end$

References

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Abelian varieties
Riemann surfaces
Mathematical identities
Theta functions