Fanography

1-15 1-17

Fano variety 1-16

hypersurface of degree 2 in $\mathbb{P}^4$

Picard rank

1 (others)

$-\mathrm{K}_X^3$

54

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

0

Mori–Mukai $d$

6

The holomorphic Poisson structures form the following irreducible components of $\mathbb{P}\mathrm{H}^0(X,\wedge^2\mathrm{T}_X)$:

See Loray–Pereira–Touzet.

index

3

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

30

$-\mathrm{K}_X$ 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 primitive.

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

• 2-7, in a curve of genus 5
• 2-13, in a curve of genus 2
• 2-17, in a curve of genus 1
• 2-21, in a curve of genus 0
• 2-23, in a curve of genus 1
• 2-26, in a curve of genus 0
• 2-29, in a curve of genus 0
• 2-31, in a curve of genus 0

This variety is fibre-like, i.e. it can appear as the fibre of a Mori fibre space.

number of moduli

0

Bott vanishing

does not hold

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

GRDB

#3

Fanosearch

#4

The naive atomic decomposition has atom dimensions $1+1+1+1$.

A standard semiorthogonal decomposition is $\langle \mathcal{S}, \mathcal{O}_X, \mathcal{O}_X(1), \mathcal{O}_X(2) \rangle$.

Both decompositions are recorded in Böhning–Graf von Bothmer–Su'a.

A full exceptional collection was constructed by Kapranov in 1986, see [MR0847146] .

Generic semisimplicity of:

• small quantum cohomology, proved by someone in at some point, see [?] , using

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

variety

$\mathbb{P}^4$

bundle

$\mathcal{O}(2)$

See the big table for more information.

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

• 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.