Hilbert schemes of elliptic surfaces: group actions and derived categories

TL;DR

This paper explores the structure of Hilbert schemes of elliptic surfaces, constructing group actions and autoequivalences of derived categories, revealing deep symmetries and extending known correspondences in algebraic geometry.

Contribution

It introduces a new group scheme acting on Hilbert schemes of elliptic surfaces and constructs autoequivalences of their derived categories using the derived McKay correspondence.

Findings

Constructed a commutative group scheme over the base of the Hilbert scheme.

Extended the group action to the entire Hilbert scheme, showing $ ext{delta}$-regularity.

Established an autoequivalence of the derived category intertwining with the group action.

Abstract

Let $X\to C$ be an elliptic surface with integral fibers and a section. The Hilbert scheme $X^{[n]}$ fibers over $C^{[n]}$. We construct a commutative group scheme over the entire base $C^{[n]}$ that embeds as an open subscheme of the Hilbert scheme, such that its action on itself extends to the entirety of $X^{[n]}$. We show that the action is $\delta$-regular in the sense of Ng\^o. Using the derived McKay correspondence, we construct an exact autoequivalence of $D^b\operatorname{Coh}(X^{[n]})$ whose kernel is a maximal Cohen-Macaulay sheaf on the fiber product. We show that this Fourier-Mukai transform intertwines with our group action, i.e. theorem of the square holds. We also discuss the case without a section using the theory of Tate-Shafarevich twists.