Fanography

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Fano variety 1-11

hypersurface of degree 6 in $\mathbb{P}(1,1,1,2,3)$

Picard rank

1 (others)

$-\mathrm{K}_X^3$

8

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

21

index

2

del Pezzo of degree 1

$X\to\mathbb{P}^2$

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

7

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

no

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

yes

hyperelliptic

no

trigonal

no

This variety is not known to be unirational.

This variety is primitive.

This variety can be blown up (in a curve) to

• 2-1, in a curve of genus 1

This variety is fibre-like, i.e. it can appear as the fibre of a Mori fibre space.

number of moduli

34

Bott vanishing

does not hold

There is no period sequence associated to this Fano 3-fold.

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

$\mathbb{P}(1^3,2,3)$

bundle

$\mathcal{O}(6)$

See the big table for more information.

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

• the book