We study the enumerative geometry of rational curves on the Hilbert schemes of points of a K3 surface.
Let be a K3 surface and let be the Hilbert scheme of points of . In the case of elliptically fibered K3 surfaces , we calculate genus-0 Gromov–Witten invariants of , which count rational curves incident to two generic fibers of the induced Lagrangian fibration . The generating series of these invariants is the Fourier expansion of a power of the Jacobi theta function times a modular form, hence of a Jacobi form.
We also prove results for genus-0 Gromov–Witten invariants of for several other natural incidence conditions. In each case, the generating series is again a Jacobi form. For the proof we evaluate Gromov–Witten invariants of the Hilbert scheme of two points of , where is an elliptic curve.
Inspired by our results, we conjecture a formula for the quantum multiplication with divisor classes on with respect to primitive curve classes. The conjecture is presented in terms of natural operators acting on the Fock space of . We prove the conjecture in the first nontrivial case . As a corollary, we find that the full genus-0 Gromov–Witten theory of in primitive classes is governed by Jacobi forms.
We present two applications. A conjecture relating genus-1 invariants of to the Igusa cusp form was proposed in joint work with R Pandharipande. Our results prove the conjecture when . Finally, we present a conjectural formula for the number of hyperelliptic curves on a K3 surface passing through two general points.
"Gromov–Witten invariants of the Hilbert schemes of points of a K3 surface." Geom. Topol. 22 (1) 323 - 437, 2018. https://doi.org/10.2140/gt.2018.22.323