A new proof of the Bondal-Orlov reconstruction using Matsui spectra

In 2005, Balmer defined the ringed space Spec_⊗ T for a given tensor triangulated category, while in 2023, the second author introduced the ringed space Spec_ T for a given triangulated category. In the algebro-geometric context, these spectra provided several reconstruction theorems using derived categories. In this paper, we prove that Spec⊗XL Perf X is an open ringed subspace of Spec_ Perf X for a quasi-projective variety X. As an application, we provide a new proof of the Bondal-Orlov and Ballard reconstruction theorems in terms of these spectra. Recently, the first author introduced the Fourier-Mukai locus SpecFM Perf X for a smooth projective variety X, which is constructed by gluing Fourier-Mukai partners of X inside Spec_ Perf X. As another application of our main theorem, we demonstrate that SpecFM Perf X can be viewed as an open ringed subspace of Spec_ Perf X. As a result, we show that all the Fourier-Mukai partners of an abelian variety X can be reconstructed by topologically identifying the Fourier-Mukai locus within Spec_ Perf X.