We prove the rationality of the descendent partition function for stable pairs on nonsingular toric 3-folds. The method uses a geometric reduction of the 2- and 3-leg descendent vertices to the 1-leg case. As a consequence, we prove the rationality of the relative stable pairs partition functions for all log Calabi-Yau geometries of the form $(X,K3)$ where $X$ is a nonsingular toric 3-fold.
Rahul PANDHARIPANDE. Aaron PIXTON. "Descendent theory for stable pairs on toric 3-folds." J. Math. Soc. Japan 65 (4) 1337 - 1372, October, 2013. https://doi.org/10.2969/jmsj/06541337