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

Derived categories and the genus of space curves