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

Abstract: 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.