Fay's trisecant identity

Template:Short description In algebraic geometry, Fay's trisecant identity is an identity between theta functions of Riemann surfaces introduced by Template:Harvs. 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 Template:Harvtxt, 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 ${ℂ}^g$ to $ℂ$
• $E$ is a prime form on $C\times C$
• $u$, $v$, $x$, $y$ are points of $C$
• $z$ is an element of ${ℂ}^g$
• $\omega$ is a 1-form on $C$ with values in ${ℂ}^g$

The Fay's identity states that

$\begin{array}{cc}& E(x,v)E(u,y)\theta (z+\underset{u}{\overset{x}{\int }}\omega )\theta (z+\underset{v}{\overset{y}{\int }}\omega )\\ -& E(x,u)E(v,y)\theta (z+\underset{v}{\overset{x}{\int }}\omega )\theta (z+\underset{u}{\overset{y}{\int }}\omega )\\ =& E(x,y)E(u,v)\theta (z)\theta (z+\underset{u+v}{\overset{x+y}{\int }}\omega )\end{array}$

with

$\begin{array}{cc}& \underset{u+v}{\overset{x+y}{\int }}\omega =\underset{u}{\overset{x}{\int }}\omega +\underset{v}{\overset{y}{\int }}\omega =\underset{u}{\overset{y}{\int }}\omega +\underset{v}{\overset{x}{\int }}\omega \end{array}$

References

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