On the Hilbert-Chow crepant resolution conjecture

TL;DR

The paper settles the Hilbert-Chow crepant resolution conjecture for all projective surfaces in exceptional curve classes by relating Hilbert schemes and orbifold symmetric products through Fulton–MacPherson interpolations. It develops a robust wall-crossing framework via epsilon-weighted and master-space constructions to reduce the problem to the affine plane, where the conjecture was previously established, and derives a DT/GW correspondence for product threefolds $S\times C$ in the zero-first-factor class. The core toolkit combines Nakajima–Grojnowski and orbifold cohomology, interpolating moduli spaces, and localisation with $I$-functions to express GW invariants across chambers, culminating in the main equality after specializing $q$ to $-1$. The work also opens avenues for generalisations, including torus-equivariant refinements and potential Chow-level lifts, and it discusses deeper links between Hilbert schemes and Fulton–MacPherson geometry beyond the considered scope.

Abstract

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.

Theorems & Definitions (34)

• Theorem 1
• Theorem 2
• Theorem 3: NePi
• Theorem 4: N22
• Corollary 1
• Definition 1
• Remark 1
• Definition 2
• Lemma 1
• proof