Let $ S $ be a K3 surface. We study the reduced Donaldson-Thomas theory of the cap
$(S\times\mathbb {P}^ 1)/S_ {\infty} $ by a second cosection argument. We obtain four main
results:(i) A multiple cover formula for the rank 1 Donaldson-Thomas theory of $\mathrm
{K3}\times E $, leading to a complete solution of this theory.(ii) Evaluation of the wall-
crossing term in Nesterov's quasi-map wallcrossing between the punctual Hilbert schemes
and Donaldson-Thomas theory of $\mathrm {K3}\times\text {Curve} $.(iii) A multiple cover …

Let S be a K3 surface. We study the reduced Donaldson–Thomas theory of the cap (S× ℙ
1)∕ S∞ by a second cosection argument. We obtain four main results:(i) A multiple cover
formula for the rank 1 Donaldson–Thomas theory of S× E, leading to a complete solution of
this theory.(ii) Evaluation of the wall-crossing term in Nesterov's quasimap wall-crossing
between the punctual Hilbert schemes and Donaldson–Thomas theory of S× Curve⁡.(iii) A
multiple cover formula for the genus 0 Gromov–Witten theory of punctual Hilbert …