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

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 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

There exist interesting semiorthogonal decompositions, but this data is not yet added.

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.

The Hilbert scheme of conics is an abelian surface.

Its Hodge diamond is