On the Hilbert-Chow crepant resolution conjecture

TL;DR

This paper proves the Hilbert-Chow crepant resolution conjecture for all projective surfaces and genera, confirming Ruan's conjecture, and explores implications for Donaldson-Thomas and Gromov-Witten theories in threefolds.

Contribution

It establishes the conjecture in exceptional curve classes for all surfaces and genera, utilizing Fulton-MacPherson compactifications to reduce the problem.

Findings

Proof of the conjecture in all projective surfaces and genera.

Confirmation of Ruan's cohomological conjecture.

Derivation of the DT/GW correspondence for specific threefold classes.

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.