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

TL;DR

This paper explores the classification of certain Q-Fano 3-folds with Gorenstein index 2 using derived categories, non-commutative projections, and Sarkisov links, offering new methods and insights into their geometric structure.

Contribution

It introduces a novel approach to classify Q-Fano 3-folds via derived category formulas and non-commutative projections, expanding on Takagi's work with new techniques.

Findings

Derived category formulas for weighted blow-ups and Kawamata blow-ups.

Behavior of derived categories under Sarkisov links analyzed.

Exceptional collections constructed with geometric interpretations.

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.