The log-local correspondence for nef log Calabi–Yau surface pairs

Let $(X,D)$ be a nef log Calabi–Yau surface pair, with $D=D_1+\cdots+D_l$, and let $d$ be a curve class. Denote by $N_{0,d}^{\rm \log}(X,D)$ and $N_{0,d}^{\rm loc}(X,D)$ the genus-zero log and local invariants, respectively. The log-local correspondence.

$$N_{0,d}^{\rm \log}(X,D) = \Bigg[ \prod_{i=1}^{l} (-1)^{d \cdot D_i+1} (d \cdot D_i)\Bigg] N_{0,d}^{\rm loc}(X,D).$$

This conjecture predicts a universal power-law relation between genus-zero log and local invariants for nef log Calabi–Yau surface pairs, extending the observed formulas in examples such as $(\mathbb{C}\mathbb{P}^2,H\cup Q)$.