2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/132409We 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.17 pages; final version, to appear in J. Reine Angew. Math.; no significant changes, only some small corrections and additions with respect to the previous versionAlgebraic Geometry14H42; 14H40text