Hilbert schemes of elliptic surfaces: Group actions and derived categories

Abstract

Let be an elliptic surface with integral fibers and a section. The Hilbert scheme fibers over . We construct a commutative group scheme over the entire base that embeds as an open subscheme of the Hilbert scheme, such that its action on itself extends to the entirety of . We show that the action is ‐regular in the sense of Ngô. Using the derived McKay correspondence, we construct an exact autoequivalence of , whose kernel is a maximal Cohen–Macaulay sheaf on the fiber product. We show that this Fourier–Mukai transform intertwines with our group action, that is, theorem of the square holds. We also discuss the case without a section using the theory of Tate‐Shafarevich twists.