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Abstract

We study the full stable pair theory --- with descendents --- of the
Calabi-Yau 3-fold $X=K_S$, where $S$ is a surface with a smooth canonical
divisor $C$.
By both $\mathbb C^*$-localisation and cosection localisation we reduce to
stable pairs supported on thickenings of $C$ indexed by partitions. We show
that only strict partitions contribute, and give a complete calculation for
length-1 partitions. The result is a surprisingly simple closed product formula
for these "vertical" thickenings.
This gives all contributions for the curve classes $[C]$ and $2[C]$ (and
those which are not an integer multiple of the canonical class). Here the
result verifies, via the descendent-MNOP correspondence, a conjecture of
Maulik-Pandharipande, as well as various results about the Gromov-Witten theory
of $S$ and spin Hurwitz numbers.