  We generalize a classical result about the genus of curves in projective
space by Gruson and Peskine to principally polarized abelian threefolds of
Picard rank one. The proof is based on wall-crossing techniques for ideal
sheaves of curves in the derived category. In the process, we obtain bounds for
Chern characters of other stable objects such as rank two sheaves. The argument
gives a proof for projective space as well. In this case these techniques also
indicate an approach for a conjecture by Hartshorne and Hirschowitz and we
prove first steps towards it.
