Derived Blow-ups and Birational Geometry of Nested Quiver Varieties

Given a quiver, Nakajima introduced the quiver variety and the Hecke correspondence, which is a closed subvariety of Cartesian products of quiver varieties. In this paper, we consider two nested quiver varieties as fiber products of Hecke correspondences along natural projections. After blowing up the diagonal, we prove that they are isomorphic to a quadruple moduli space which Negu\c{t} observed for the Jordan quiver. The main ingredient is Jiang's derived projectivization theory and Hekking's derived blow-up theory, while we obtain the local model when both the two derived schemes are quasi-smooth.