Fanography

2-25 2-27

Fano variety 2-26

blowup of 1-15 in a line

Picard rank

2 (others)

$-\mathrm{K}_X^3$

34

$\mathrm{h}^{1,2}(X)$

0

Mori–Mukai $d$

-21

Still open.

index

1

$\dim\mathrm{H}^0(X,\omega_X^\vee)$

20

$-\mathrm{K}_X$ very ample?

yes

$-\mathrm{K}_X$ basepoint free?

yes

hyperelliptic

no

trigonal

no

This variety is rational.

This variety is stably rational.

The Weyl group associated to the KKMR decomposition is $\{e\}$; see the Weyl-group table for sources and conventions.

This variety is the blowup of

• 1-15, in a curve of genus 0
• 1-16, in a curve of genus 0

number of moduli

0

Bott vanishing

holds

The following period sequences are associated to this Fano 3-fold:

GRDB

#58

Fanosearch

#47

A full exceptional collection can be constructed using Orlov's blowup formula.

Generic semisimplicity of:

• small quantum cohomology, proved by Ciolli in 2005, see [MR2168069] , using the description of a 1 or 2-curve blowup of $\mathbb{P}^3$ or $Q^3$

Fano 3-folds from homogeneous vector bundles over Grassmannians gives the following description(s):

variety

$\operatorname{Gr}(2,4) \times \operatorname{Gr}(2,5)$

bundle

$\mathcal{Q}_{\operatorname{Gr}(2,4)} \boxtimes \mathcal{U}_{\operatorname{Gr}(2,5)}^{\vee} \oplus \mathcal{O}(1,0) \oplus \mathcal{O}(0,1)^{\oplus 2}$

variety

$\operatorname{Fl}(2,3,5)$

bundle

$\mathcal{U}_1^{\vee} \oplus \mathcal{O}(1,0) \oplus \mathcal{O}(0,1)^{\oplus 2}$

See the big table for more information.

• none are K-stable
• none are K-polystable
• general member is K‑semistable but there exist members that are not

• the book

For this deformation family, every smooth member has coregularity 0.

The generic and uniform statements are due to Avilov–Loginov–Przyjalkowski; the existence of a smooth member of coregularity zero in every family is due to Zhakupov.

See coregularity for terminology.

For every K-polystable member, the Futaki invariant vanishes identically on the entire Kähler cone.

This is the computation of Sektnan–Tipler.