Derived hyperquot schemes
Sergej Monavari, Emanuele Pavia, Andrea T. Ricolfi
TL;DR
The paper constructs a derived enhancement $\mathbf{R}\mathrm{Quot}_X^{[l]}({\mathcal{E}})$ of the classical hyperquot scheme for a projective $X$ and a perfect coherent sheaf ${\mathcal{E}}$, and computes its global tangent complex in terms of universal flags of perfect complexes. It then derives a natural obstruction theory on the classical hyperquot scheme $\mathrm{Quot}_X^{[l]}({\mathcal{E}})$, which in the curve case yields a virtual fundamental class consistent with prior work. The approach uses flags of perfect complexes, derived stacks, and the functoriality of cotangent/tangent complexes to connect derived geometry with classical moduli theory. Amplifications extend the constructions to quasiprojective $X$ and arbitrary bases via pseudo-perfect setups and $E_{\infty}$-ring frameworks, situating the results within broader derived-Quot and derived-Grassmannian literature and enabling broader applicability to enumerative geometry and deformation theory.
Abstract
We define a derived enhancement of the hyperquot scheme (also known as nested Quot scheme), which classically parametrises flags of quotients of a perfect coherent sheaf on a projective scheme. We prove it is representable by a derived scheme, and we compute its global tangent complex. As an application, we provide a natural obstruction theory on the classical hyperquot scheme. The latter recovers the virtual fundamental class recently constructed by the first and third author in the context of the enumerative geometry of hyperquot schemes on smooth projective curves.