algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

, Fay's trisecant identity is an identity between

theta function In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field theo ...

s of

Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...

s introduced by . Fay's identity holds for theta functions of Jacobians of curves, but not for theta functions of general

abelian varieties In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular function ...

. The name "trisecant identity" refers to the geometric interpretation given by , who used it to show that the

Kummer variety In mathematics, the Kummer variety of an abelian variety is its quotient by the map taking any element to its inverse. The Kummer variety of a 2-dimensional abelian variety is called a Kummer surface In algebraic geometry, a Kummer quartic su ...

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'' *θ is the Riemann theta function of ''C'', a function from C^''g'' to C *''E'' is a

prime form In algebraic geometry, the Schottky–Klein prime form ''E''(''x'',''y'') of a compact Riemann surface ''X'' depends on two elements ''x'' and ''y'' of ''X'', and vanishes if and only if ''x'' = ''y''. The prime form ''E'' is not quite ...

on ''C''×''C'' *''u'',''v'',''x'',''y'' are points of ''C'' *''z'' is an element of C^''g'' *ω is a 1-form on ''C'' with values in C^''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

* * * {{Authority control Abelian varieties Riemann surfaces Mathematical identities Theta functions