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Let $S$ be a K3 surface. We study the reduced Donaldson-Thomas theory of thecap $(S \times \mathbb{P}^1) / S_{\infty}$ by a second cosection argument. Weobtain four main results: (i) A multiple cover formula for the rank 1Donaldson-Thomas theory of $\mathrm{K3} \times E$, leading to a completesolution of this theory. (ii) Evaluation of the wall-crossing term inNesterov's quasi-map wallcrossing between the punctual Hilbert schemes andDonaldson-Thomas theory of $\mathrm{K3} \times \text{Curve}$. (iii) A multiplecover formula for the genus $0$ Gromov-Witten theory of punctual Hilbertschemes. (iv) Explicit evaluations of virtual Euler numbers of Quot schemes ofstable sheaves on K3 surfaces.

Multiple cover formulas for K3 geometries, wall-crossing, and Quot schemes