Gluing of Fourier-Mukai partners in a triangular spectrum and birational geometry
Daigo Ito
TL;DR
The paper introduces the Fourier-Mukai locus Spec^{FM} T inside Matsui's triangular spectrum, capturing all Fourier-Mukai partners X with Perf X ≅ T as glued tt-spectra from each partner. It proves Spec^{FM} T is a smooth scheme of finite type, with each copy of X embedded as an open subscheme and dimension matching dim X, and analyzes how overlaps encode birational relations among FM partners. Through extensive examples—abelian varieties, spherical twists on surfaces, toric varieties, and flops/K-equivalences—it demonstrates how geometric and birational properties manifest in the gluing of tt-spectra and discusses conjectural, categorically flavored refinements like the tt-DK hypothesis and Serre-invariant loci. The framework provides a structural, scheme-theoretic perspective on FM partners, enabling refined comparisons and potential categorical pathways to birational geometry and K-equivalence. Overall, the work links derived-category invariants to birational phenomena via a coherent gluing of tt-spectra, offering a new tool for understanding FM partners and their geometric implications.
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.