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 induced by theta functions of order 2, has a 4-dimensional space of trisecants.

Statement

Suppose that

• is a compact Riemann surface
• is the genus of
• is the Riemann theta function of , a function from to
• is a prime form on
• , , , are points of
• is an element of
• is a 1-form on with values in

The Fay's identity states that

with

References