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

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 $H_{\ast }({\mathbb{N}}^{\text{pl}},\text{Q})$ of the projective-linear pairs stack, and next pass to Gross’s polynomial realization $e^{\kappa }\text{Q}[s_{jk\mathrm{\ell }}]$. 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 ${\text{PP}}^3$, on a smooth cubic threefold, on ${\text{Bl}}_p{\text{PP}}^3$, on ${\text{Bl}}_{\mathrm{\ell }}{\text{PP}}^3$, and on the projective-bundle threefold $\text{PP}({\text{OO}}_X\oplus {\text{OO}}_X(-1,-1))$ over $X={\text{PP}}^1\times {\text{PP}}^1$. In the ${\text{PP}}^3$ 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.