  We prove and conjecture results which show that Castelnuovo theory in
projective space has a close analogue for abelian varieties. This is related to
the geometric Schottky problem: our main result is that a principally polarized
abelian variety satisfies a precise version of the Castelnuovo Lemma if and
only if it is a Jacobian. This result has a surprising connection to the
Trisecant Conjecture. We also give a genus bound for curves in abelian
varieties.
