Fanography

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Fano variety 1-9

section of the adjoint $\mathrm{G}_2$-Grassmannian $\mathrm{G}_2\mathrm{Gr}(2,7)$ by codimension 2 subspace

Picard rank

1 (others)

$-\mathrm{K}_X^3$

18

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

2

Mori–Mukai $d$

18

No nontrivial holomorphic Poisson structures.

See Loray–Pereira–Touzet.

index

1

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

12

$-\mathrm{K}_X$ 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

10

Bott vanishing

does not hold

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

GRDB

#124

Fanosearch

#10

The naive atomic decomposition has atom dimensions $2+1+1$.

A standard semiorthogonal decomposition is $\langle \mathrm{D}^{\mathrm{b}}(C_2), \mathcal{E}_2, \mathcal{O}_X \rangle$.

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

By Hertling–Manin–Teleman we have that quantum cohomology cannot be generically semisimple, as $\mathrm{h}^{1,2}\neq 0$.

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

variety

$\operatorname{Gr}(2,7)$

bundle

$\mathcal{Q}^{\vee}(1) \oplus \mathcal{O}(1)^{\oplus2}$

See the big table for more information.

• general member is K‑stable
• general member is K‑polystable
• 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 an abelian surface.

Its Hodge diamond is