On the Hilbert-Chow crepant resolution conjecture

We prove the Hilbert-Chow crepant resolution conjecture in the exceptional curve classes for all projective surfaces and all genera. In particular, this confirms Ruan's cohomological Hilbert-Chow crepant resolution conjecture. The proof exploits Fulton-MacPherson compactifications, reducing the conjecture to the case of the affine plane. As an application, using previous results of the author, we also deduce the families DT/GW correspondence for threefolds $S \times C$ in classes that are zero on the first factor, yielding a wall-crossing proof of the correspondence in this case. Finally, we speculate on the relationship between Hilbert schemes and Fulton-MacPherson compactifications beyond the topics considered in this work.