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Fano fibrations and DK conjecture for relative Grassmann flips

Marco Rampazzo

Abstract

Given a vector bundle $\mathcal E$ on a smooth projective variety $B$, the flag bundle $\mathcal F l(1,2,\mathcal E)$ admits two projective bundle structures over the Grassmann bundles $\mathcal G r(1, \mathcal E)$ and $G r(2, \mathcal E)$. The data of a general section of a suitably defined line bundle on $\mathcal F l(1,2,\mathcal E)$ defines two varieties: a cover $X_1$ of $B$ and a fibration $X_2$ on $B$ with general fiber isomorphic to a smooth Fano variety. We construct a semiorthogonal decomposition of the derived category of $X_2$ which consists of a list of exceptional objects and a subcategory equivalent to the derived category of $X_1$. As a byproduct, we obtain a new full exceptional collection for the Fano fourfold of degree $12$ and genus $7$. Any birational map of smooth projective varieties which is resolved by blowups with exceptional divisor $\mathcal F l(1, 2, \mathcal E)$ is an instance of a so-called Grassmann flip: we prove that the DK conjecture of Bondal-Orlov and Kawamata holds for such flips. This generalizes a previous result of Leung and Xie to a relative setting.

Fano fibrations and DK conjecture for relative Grassmann flips

Abstract

Given a vector bundle on a smooth projective variety , the flag bundle admits two projective bundle structures over the Grassmann bundles and . The data of a general section of a suitably defined line bundle on defines two varieties: a cover of and a fibration on with general fiber isomorphic to a smooth Fano variety. We construct a semiorthogonal decomposition of the derived category of which consists of a list of exceptional objects and a subcategory equivalent to the derived category of . As a byproduct, we obtain a new full exceptional collection for the Fano fourfold of degree and genus . Any birational map of smooth projective varieties which is resolved by blowups with exceptional divisor is an instance of a so-called Grassmann flip: we prove that the DK conjecture of Bondal-Orlov and Kawamata holds for such flips. This generalizes a previous result of Leung and Xie to a relative setting.
Paper Structure (26 sections, 17 theorems, 84 equations)

This paper contains 26 sections, 17 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Flag bundles and fibrations
  3. Notation
  4. The flag bundle
  5. Two fibrations over something
  6. A semiorthogonal decomposition for something
  7. Two semiorthogonal decompositions for something
  8. Extending bundles from the fibers
  9. The chess game - rules
  10. Mutations of blocks
  11. Arrangement of boxes and semiorthogonal decompositions
  12. The chess game - mutations for the even case
  13. First upward phase
  14. Second upward phase
  15. The chess game - mutations for the odd case
  16. ...and 11 more sections

Key Result

Theorem 1.2

Consider a Grassmann flip $\mu:{\mathcal{X}}_1\dashrightarrow{\mathcal{X}}_2$ such that the exceptional divisor of the flip is isomorphic to a flag bundle ${\mathcal{F}} l(1,2,{\mathcal{E}})$ over a smooth projective base. Then $D^b({\mathcal{X}}_1)\subset D^b({\mathcal{X}}_2)$, i.e. $\mu$ satisfies

Theorems & Definitions (35)

  • Conjecture 1.1: DK conjecture
  • Theorem 1.2
  • Theorem 1.3
  • Remark 2.1
  • Lemma 2.5
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 25 more