Edited 4θ-embeddings of Jacobians

By the Lefschetz embedding theorem a principally polarized g-dimensional abelian variety is embedded into projective space by the linear system of 4 half-characteristic theta functions. Suppose we edit this linear system by dropping all the theta functions vanishing at the origin to order greater than parity requires. We prove that for Jacobians the edited 4Θ linear system still defines an embedding into projective space. Moreover, we prove that the projective models of Jacobians arising from the elementary algebraic construction of Jacobians recently given by the author are (after passage to linear hulls) copies of the edited 4Θ model. We obtain our results by aptly combining the quartic and determinantal identities satisfied by the Riemann theta function. We take the somewhat nonstandard tack of working in the framework of Weil’s old book on Kähler varieties in order to avoid having to make extremely complicated calculations.