Hilbert schemes of elliptic surfaces: group actions and derived categories
David Zhiyuan Bai
TL;DR
The paper studies Hilbert schemes $X^{[n]}$ of an elliptic surface $\pi:X\to C$ with integral fibers and a section, introducing a natural group scheme $X^{[n],\pi}$ over $C^{[n]}$ that acts on $X^{[n]}$ and forms a $\delta$-regular abelian fibration. Using the derived McKay correspondence, it constructs a maximal Cohen–Macaulay kernel $\mathcal{P}_n^{\mathrm{BKR}}$ from which an exact autoequivalence of $D^b\operatorname{Coh}(X^{[n]})$ arises, and proves a Theorem of the square that intertwines this action with the kernel. The construction relies on the Bridgeland–King–Reid equivalence and Pløog’s outer tensor framework, and it naturally extends to the no-section case via Tate–Shafarevich twists, yielding equivalences with twisted derived categories $D^b\operatorname{Coh}(X^{[n]},\alpha)$. Altogether, the work provides a robust autoduality and group-action framework for Hilbert schemes in the elliptic-surface setting and connects these to twisted derived categories in the absence of a section.
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 $δ$-regular in the sense of Ngô. 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.