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

Xingbang Hao

TL;DR

The paper develops a derived-category framework to classify $ olinebreak[0] extcolor{red}{ ext{Q-Fano}}$ 3-folds of Gorenstein index $2$ within Takagi’s hourglass paradigm, by analyzing how weighted Kawamata and Francia-type birational moves affect $D^b$ via semi-orthogonal decompositions and non-commutative projections. It constructs explicit formulas for the derived category through weighted blow-ups, Kawamata blow-ups, and Francia flips, and shows that Takagi hourglasses admit residue categories governed by matrix-factorization categories, with a concrete cubic-type example $X_{1.9}$ yielding an isomorphism of residue MF-categories with $ ext{MF}(B_3)$. The main contributions include (i) a general weighted-blow-up derived-category formula, (ii) a framework for non-commutative hourglasses and their exceptional collections, and (iii) an explicit instance demonstrating MF-equivalence to a classical cubic model, supporting a geometric reading of the non-commutative components. These results advance categorical methods in birational classification of low-dimensional Fano varieties and hint at deeper connections between Sarkisov links and non-commutative invariants in algebraic geometry.

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.

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

TL;DR

The paper develops a derived-category framework to classify 3-folds of Gorenstein index within Takagi’s hourglass paradigm, by analyzing how weighted Kawamata and Francia-type birational moves affect via semi-orthogonal decompositions and non-commutative projections. It constructs explicit formulas for the derived category through weighted blow-ups, Kawamata blow-ups, and Francia flips, and shows that Takagi hourglasses admit residue categories governed by matrix-factorization categories, with a concrete cubic-type example yielding an isomorphism of residue MF-categories with . The main contributions include (i) a general weighted-blow-up derived-category formula, (ii) a framework for non-commutative hourglasses and their exceptional collections, and (iii) an explicit instance demonstrating MF-equivalence to a classical cubic model, supporting a geometric reading of the non-commutative components. These results advance categorical methods in birational classification of low-dimensional Fano varieties and hint at deeper connections between Sarkisov links and non-commutative invariants in algebraic geometry.

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 -Fano 3-fold with Fano index 1/2 and at worst 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.

Paper Structure

This paper contains 21 sections, 65 theorems, 394 equations, 1 table.

Table of Contents

  1. Introduction
  2. Notations
  3. Preliminaries
  4. Stack elementary
  5. Weighted blowing up
  6. Smooth type
  7. Kawamata type
  8. Semi-orthogonal decomposition
  9. Derived category
  10. Derived category of blowing up
  11. Dual collection on weighted projective space
  12. Derived category of weighted blow-up
  13. Derived category of Kawamata blow-up
  14. Derived category of Francia's flip
  15. Takagi's hourglasses
  16. ...and 6 more sections

Key Result

Theorem 1.1

If $X$ is a smooth variety, after $(a_{1},..,a_{c})$A sequence of non-negative integers has no non-trivial common divisor no:ai.-weighted blowing up at a codimension $c$ smooth center $S$, we get another variety (algebraic space) $Y$ with only cyclic quotient singularities and we denote its canonica where $\mathrm{\Psi}$ and $\mathrm{\Psi}_{i}$ are defined by $\mathrm{\Psi}(-):=\pi^{*}(-)$ and $\m

Theorems & Definitions (133)

  • Theorem 1.1: KalogKaTor, Thm. \ref{['thm:smoohtweightedblowup']}
  • Theorem 1.2: Thm. \ref{['thm:singularweightedblowup']}
  • Proposition 1.1: Prop. \ref{['prop:pullbackcenterlocal']}
  • Proposition 1.2: Prop. \ref{['prop:secondshoukufr']}
  • Definition 1.1: Tak Main Assumption 0.1
  • Theorem 1.3: Thm. \ref{['thm:SODh']}
  • Proposition 1.3: Prop. \ref{['prop:X1.9']}
  • Proposition 1.4: Non-commutative projection, Prop. \ref{['prop:projection']}-\ref{['prop:sgequi']}
  • Corollary 1.3.1
  • Definition 3.1
  • ...and 123 more