Gromov–Witten theory of Hilbert schemes ofpoints on elliptic surfaces with multiple fibers

We study the Gromov-Witten theory of Hilbert schemes of points on elliptic surfaces with multiple fibers.We prove a vanishing theorem for the Gromov-Witten invariants of these Hilbert schemes, and compute the exceptional genus-0 case for the Hilbert schemes of two points on elliptic surfaces with exactly one multiple fiber.The strategy is to use the theory of cosection localization and compute a certain obstruction sheaf.