Fanography

1-9 1-11

Fano variety 1-10

zero locus of $(\bigwedge^2\mathcal{U}^\vee)^{\oplus 3}$ on $\mathrm{Gr}(3,7)$

Picard rank

1 (others)

$-\mathrm{K}_X^3$

22

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

0

Mori–Mukai $d$

22

Remarks

The unique such threefold where $\mathrm{Aut}(X)\cong\mathrm{PGL}_2$ is called the Mukai–Umemura 3-fold

The generic member admits no nontrivial holomorphic Poisson structure; the special Mukai–Umemura member admits a 1-dimensional family (type $\mathrm{Aff}$).

See Loray–Pereira–Touzet.

index

1

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

14

$-\mathrm{K}_X$ 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 is fibre-like, i.e. it can appear as the fibre of a Mori fibre space.

number of moduli

6

Bott vanishing

does not hold

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

GRDB

#113

Fanosearch

#17

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

A standard semiorthogonal decomposition is $\langle \mathcal{E}_4, \mathcal{E}_3, \mathcal{E}_2, \mathcal{O}_X \rangle$.

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

A full exceptional collection was constructed by Kuznetsov in 1996, see [MR1445274] .

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

$\operatorname{Gr}(3,7)$

bundle

$(\bigwedge^2\mathcal{U}^\vee)^{\oplus 3}$

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
• general member is K‑semistable

• the book

For this deformation family, the general 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.

The Hilbert scheme of conics is $\mathbb{P}^2$.

Its Hodge diamond is