  The elliptic genus of K3 is an index for the 1/4-BPS states of its sigma
model. At the torus orbifold point there is an accidental degeneracy of such
states. We blow up the orbifold fixed points using conformal perturbation
theory, and find that this fully lifts the accidental degeneracy of the 1/4-BPS
states with h=1. At a generic point near the Kummer surface the elliptic genus
thus measures not just their index, but counts the actual number of these BPS
states. We comment on the implication of this for symmetry surfing and Mathieu
moonshine.
