Interpolation of fat points on K3 and abelian surfaces

We prove that any number of general fat points of any multiplicities impose the expected number of conditions on a linear system on a smooth projective surface, in several cases including primitive linear systems on very general K3 and abelian surfaces, “Du Val” linear systems on blowups of at nine very general points, and certain linear systems on some ruled surfaces over elliptic curves. This is done by answering a question of the author about the case of only one fat point on a certain ruled surface, which follows from a circle of results due to Treibich–Verdier, Segal–Wilson, and others.