Derived Categories of Derived Grassmannians

TL;DR

This paper develops semiorthogonal decompositions for derived Grassmannians of perfect complexes, confirming a conjecture and extending previous results, with applications to classical algebraic geometry problems.

Contribution

It introduces new semiorthogonal decompositions for derived Grassmannians of perfect complexes, verifying a conjecture and generalizing Toda's work using derived algebraic geometry.

Findings

Verified the Quot formula conjecture for derived Grassmannians.

Generalized Toda's semiorthogonal decomposition results.

Provided applications to blowups, reducible schemes, and linear series.

Abstract

This paper establishes semiorthogonal decompositions for derived Grassmannians of perfect complexes with Tor-amplitude in $[0,1]$. This result verifies the author's Quot formula conjecture [J21a] and generalizes and strengthens Toda's result in [Tod23]. We give applications of this result to various classical situations such as blowups of determinantal ideals, reducible schemes, and varieties of linear series on curves. Our approach utilizes the framework of derived algebraic geometry, allowing us to work over arbitrary base spaces over $\mathbb{Q}$. It also provides concrete descriptions of Fourier-Mukai kernels in terms of derived Schur functors.