  We study twisted ideal sheaves of small length on an irreducible principally
polarized abelian surface (T,l). Using Fourier-Mukai techniques we associate
certain jumping schemes to such sheaves and completely classify such loci. We
give examples of applications to the enumerative geometry of T and show that no
smooth genus 5 curve on such a surface can contain a g^1_3. We also describe
explicitly the singular divisors in the linear system |2l|.
