Remarks on GW/PT under del Pezzo transitions

Abstract

A projective threefold transition $Y\stackrel{\varphi }{\to }\bar{Y}⇝X$ is del Pezzo if $\varphi$ 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.

arXiv comment16 pages, Comments are welcome