Summary: EDITED 4-EMBEDDINGS OF JACOBIANS
GREG W. ANDERSON
Abstract. By the Lefschetz embedding theorem a principally po-
larized g-dimensional abelian variety is embedded into projective
space by the linear system of 4g
half-characteristic theta functions.
Suppose we edit this linear system by dropping all the theta func-
tions vanishing at the origin to order greater than parity requires.
We prove that for Jacobians the edited 4 linear system still de-
fines an embedding into projective space. Moreover, we prove that
the projective models of Jacobians arising from the elementary al-
gebraic construction of Jacobians recently given by the author are
(after passage to linear hulls) copies of the edited 4 model. We
obtain our results by aptly combining the quartic and determinan-
tal identities satisfied by the Riemann theta function. We take
the somewhat nonstandard tack of working in the framework of
Weil's old book on Kšahler varieties in order to avoid having to
make extremely complicated calculations.
The point of departure for this paper is the elementary algebraic con-