Fanography

3-24 3-26

Fano variety 3-25

blowup of 1-17 in the disjoint union of two lines

Alternative description:

• $\mathbb{P}(\mathcal{O}(1,0)\oplus\mathcal{O}(0,1))$ over $\mathbb{P}^1\times\mathbb{P}^1$

Picard rank

3 (others)

$-\mathrm{K}_X^3$

44

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

0

index

1

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

25

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

yes

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

yes

hyperelliptic

no

trigonal

no

This variety is rational.

This variety is the blowup of

• 2-33, in a curve of genus 0

This variety can be blown up (in a curve) to

• 4-6, in a curve of genus 0
• 4-9, 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

#16

Fanosearch

#41

$\mathbb{P}^1$-bundle over $\mathbb{P}^1\times\mathbb{P}^1$, for the vector bundle $\mathcal{O}_{\mathbb{P}^1\times\mathbb{P}^1}(1,0)\oplus\mathcal{O}_{\mathbb{P}^1\times\mathbb{P}^1}(0,1)$.

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

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

bundle

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

variety

$\operatorname{Fl}(1,2,4)$

bundle

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

See the big table for more information.

• none are K-stable
• every member is K‑polystable
• every member is K‑semistable

• the book

This variety is toric.

It corresponds to ID #18 on grdb.co.uk.