arXiv subjects

Xingbang Hao

Publications and source records attributed to Xingbang Hao.

3 recordsLinked to original sources

arXiv2026

Noncommutative Hourglasses II 1/2: Derived Category of Birational Linkage with Well-Formed Singularities

We consider varieties whose singularities are complete intersections within toric singularities, which we call well-formed varieties. Using the construction of equivariant standard resolutions, we can extend several classical formulas in the derived category of bounded coherent sheaves to this singular setting. Finally, we also discuss the Bondal--Orlov localization for certain morphisms between terminal complex threefolds and their stacks.

math.AG↗

arXiv2025

Constructing and transforming standard resolutions of local modules

We construct standard resolutions for analytic local modules on complex hypersurfaces using standard basis methods, with extensions to complete intersections. The algebraic version over arbitrary infinite fields is also suggested. Applications include derived category formula of local hypersurface blow-ups, revealing new connections between standard bases and categorical geometry.

math.AG↗

arXiv2025

Non-commutative hourglasses I: On classification of the Q-Fano 3-folds Gorenstein index 2 via Derived category

In previous work, Takagi used the methods of solving the Sarkisov links by calculating the corresponding Diophantine equations and the construction of key varieties to give all possible classifications and some implementations of a class $\mathbb{Q}$-Fano 3-fold with Fano index 1/2 and at worst $(1, 1, 1)/2$ or QODP singularities. Firstly, we use a method different from Kawamata's work to give the derived category formulas for general weighted blow-up and Kawamata weighted blow-up. On this basis, we study the changing behavior of the derived category of Takagi's varieties under Sarkisov links. Finally, by studying non-commutative projections, we give exceptional collections on the derived category of Takagi's varieties and their corresponding geometric meanings.

math.AG↗