Fanography

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Fano variety 1-5: Gushel–Mukai 3-fold

1. section of Plücker embedding of $\mathrm{Gr}(2,5)$ by codimension 2 subspace and a quadric
2. double cover of 1-15 with branch locus an anticanonical divisor

Picard rank

1 (others)

$-\mathrm{K}_X^3$

10

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

10

index

1

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

8

$-\mathrm{K}_X$ very ample?

yes if (a), else no

$-\mathrm{K}_X$ basepoint free?

yes

hyperelliptic

no

trigonal

no

This variety is known to be generically non-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

1. 22
2. 19

Bott vanishing

does not hold

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

GRDB

#160

Fanosearch

#9

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

$\operatorname{Gr}(2,5)$

bundle

$\mathcal{O}(2) \oplus \mathcal{O}(1)^{\oplus2}$

See the big table for more information.