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

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.