Derived hyperquot schemes

TL;DR

This paper introduces a derived version of the hyperquot scheme, proving its representability as a derived scheme, computing its tangent complex, and applying it to define a natural obstruction theory that recovers known virtual fundamental classes.

Contribution

It constructs a derived enhancement of the hyperquot scheme, providing new tools for its geometric and enumerative analysis.

Findings

Proves the derived hyperquot scheme is representable by a derived scheme

Computes the global tangent complex of the derived hyperquot scheme

Provides a natural obstruction theory recovering existing virtual fundamental classes

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.