Fanography

4-12 5-1

Fano variety 4-13

blowup of 3-27 in a curve of degree $(1,1,3)$

Picard rank

4 (others)

$-\mathrm{K}_X^3$

26

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

0

Mori–Mukai $d$

-44

Still open.

index

1

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

16

$-\mathrm{K}_X$ 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 $\mathrm{A}_1$; see the Weyl-group table for sources and conventions.

This variety is the blowup of

• 3-27, in a curve of genus 0
• 3-31, in a curve of genus 0

number of moduli

1

Bott vanishing

does not hold

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

GRDB

#88

Fanosearch

#32

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

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

variety

$ \mathbb{P}^1_1 \times \mathbb{P}^1_2 \times \mathbb{P}^1_3 \times \mathbb{P}^4$

bundle

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

See the big table for more information.

• general member is K‑stable but there exist members that are not
• general member is K‑polystable but there exist members that are not
• every member is K‑semistable

• the book
• 2309.12518

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.