Gluing of Fourier-Mukai partners in a triangular spectrum and birational geometry

TL;DR

This paper introduces the Fourier-Mukai (FM) locus within the triangular spectrum of a triangulated category, linking FM partners of varieties to their geometric and birational properties through spectral gluing.

Contribution

It constructs the FM locus in the triangular spectrum, providing a categorical framework to study FM partners and their geometric relations.

Findings

The FM locus is embedded in the triangular spectrum as a union of spectra of FM partners.

Geometric and birational properties are reflected in the spectral gluing within the FM locus.

Comparison with other loci suggests deep categorical connections, including a conjecture relating to the Serre invariant locus.

Abstract

Balmer defined the tensor triangulated spectrum $\operatorname{Spec}_\otimes \mathcal{T}$ of a tensor triangulated category $(\mathcal{T},\otimes)$ and showed that for a variety $X$, we have the reconstruction $X \cong \operatorname{Spec}_{\otimes_{\mathscr{O}_X}^{\mathbb{L}}}\operatorname{Perf} X$. In the absence of the tensor structure, Matsui recently introduced the triangular spectrum $\operatorname{Spec}_\vartriangle \mathcal{T}$ of a triangulated category $\mathcal{T}$ and showed that there exists an immersion $X \cong \operatorname{Spec}_{\otimes_{\mathscr{O}_X}^{\mathbb{L}}}\operatorname{Perf} X \subset \operatorname{Spec}_\vartriangle \operatorname{Perf} X$. In this paper, we construct a scheme $\operatorname{Spec}^{\mathsf{FM}} \mathcal{T} \subset \operatorname{Spec}_\vartriangle \mathcal{T}$, called the Fourier-Mukai (FM) locus, by gathering all varieties $X$ satisfying $\operatorname{Perf} X \simeq \mathcal{T}$. Those varieties are called FM partners of $\mathcal{T}$ and immersed into $\operatorname{Spec}_\vartriangle \mathcal{T}$ as tensor triangulated spectra. We present a variety of examples illustrating how geometric and birational properties of FM partners are reflected in the way their tensor triangulated spectra are glued in the FM locus. Finally, we compare the FM locus with other loci within the triangular spectrum admitting categorical characterizations, and in particular, make a precise conjecture about the relation of the FM locus with the Serre invariant locus.