Fanography

5-1 5-3

Fano variety 5-2

blowup of 3-25 in the disjoint union of two exceptional lines on the same irreducible component

Picard rank

5 (others)

$-\mathrm{K}_X^3$

36

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

0

index

1

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

21

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

yes

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

yes

hyperelliptic

no

trigonal

no

This variety is rational.

This variety is the blowup of

• 4-9, in a curve of genus 0
• 4-11, in a curve of genus 0
• 4-12, 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

#64

Fanosearch

#87

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

Alternatively, Kawamata has constructed a full exceptional collection for every smooth projective toric variety.

Generic semisimplicity of:

• small quantum cohomology, proved by Iritani in 2007, see [MR2359850] , using toric geometry

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

variety

$\mathbb{P}^1 \times \mathbb{P}^3 \times \mathbb{P}^{8} \times \mathbb{P}^1$

bundle

$\mathcal{O}(1,1,0,0) \oplus \Lambda(0,0,1,0) \oplus \mathcal{O}(-1,1,1,0) \oplus \mathcal{O}(0,1,0,1)$

See the big table for more information.

• none are K-stable
• none are K-polystable
• none are K-semistable

• the book

This variety is toric.

It corresponds to ID #9 on grdb.co.uk.