Fanography

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Fano variety 1-8

section of Plücker embedding of $\mathrm{SGr}(3,6)$ by codimension 3 subspace

Picard rank

1 (others)

$-\mathrm{K}_X^3$

16

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

3

Mori–Mukai $d$

16

No nontrivial holomorphic Poisson structures.

See Loray–Pereira–Touzet.

index

1

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

11

$-\mathrm{K}_X$ 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

12

Bott vanishing

does not hold

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

GRDB

#143

Fanosearch

#20

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

$\operatorname{Gr}(3,6)$

bundle

$\bigwedge^2\mathcal{U}^{\vee} \oplus \mathcal{O}(1)^{\oplus3}$

See the big table for more information.

• every member is K‑stable
• every member is K‑polystable
• every member is K‑semistable

• the book

The Hilbert scheme of conics is the ruled surface obtained from projectivisation of simple rank 2 bundle over a smooth curve of genus 3.

Its Hodge diamond is