Examples of descendent generating series for Pandharipande--Thomas stable pairs on smooth projective Fano threefolds via one-dimensional wall-crossing

Abstract

We study descendent generating series for Pandharipande--Thomas stable pairs on smooth projective Fano threefolds. We use the wall-crossing setup developed by the author and Joyce in Joyce's Lie algebra $H_*(\N^{\pl},\Q)$ of the projective-linear pairs stack, and next pass to Gross's polynomial realization $e^κ\Q[s_{jk\ell}]$. We compute explicit examples of one-dimensional Donaldson--Thomas invariants on Fano 3-folds and, via wall-crossing, Pandharipande--Thomas stable pair invariants and descendent generating series. We compute examples on $\PP^3$, on a smooth cubic threefold, on $\Bl_p\PP^3$, on $\Bl_\ell\PP^3$, and on the projective-bundle threefold $\PP(\OO_X\oplus \OO_X(-1,-1))$ over $X=\PP^1\times\PP^1$. In the $\PP^3$ and cubic threefold examples we compare the intrinsic large-$n$ tails with the formulas of Pandharipande and Moreira and show that, in the cases treated in common, the differences are Laurent polynomials.