Remarks on GW/PT under del Pezzo transitions

TL;DR

This paper demonstrates that the Gromov-Witten/PT correspondence persists through del Pezzo transitions in threefolds, providing new examples and utilizing degeneration and deformation techniques to relate known cases.

Contribution

It establishes the GW/PT correspondence for threefolds undergoing del Pezzo transitions, including a new example involving a degree 6 hypersurface in weighted projective space.

Findings

GW/PT correspondence holds after del Pezzo transitions.

A degree 6 hypersurface in P(3,2,1,1,1) exemplifies the correspondence.

Degeneration and deformation methods reduce complex cases to known results.

Abstract

A projective threefold transition $Y \xrightarrow{\phi} \bar{Y} \rightsquigarrow X$ is del Pezzo if $\phi$ contracts a smooth del Pezzo surface to a point. We show that the GW/PT correspondence holds on $Y$ implies that it holds on $X$. In particular, a hypersurface of degree $6$ in $\mathbb{P} (3, 2, 1, 1, 1)$ gives a new example to the correspondence. The main tools are (i) the double point degeneration constructed in arXiv:2508.01374 and (ii) deformations of del Pezzo surfaces into toric surfaces (Proposition 3.12). Applications of the degeneration formulas in GW and PT then reduce the problem to known cases.