  We consider the compactification of the E8xE8 heterotic string on a K3
surface with "the spin connection embedded in the gauge group" and the dual
picture in the type IIA string (or F-theory) on a Calabi-Yau threefold X. It
turns out that the same X arises also as dual to a heterotic compactification
on 24 point-like instantons. X is necessarily singular, and we see that this
singularity allows the Ramond-Ramond moduli on X to split into distinct
components, one containing the (dual of the heterotic) tangent bundle, while
another component contains the point-like instantons. As a practical
application we derive the result that a heterotic string compactified on the
tangent bundle of a K3 with ADE singularities acquires nonperturbatively
enhanced gauge symmetry in just the same fashion as a type IIA string on a
singular K3 surface. On a more philosophical level we discuss how it appears to
be natural to say that the heterotic string is compactified using an object in
the derived category of coherent sheaves. This is necessary to properly extend
the notion of T-duality to the heterotic string on a K3 surface.
