Fanography

1-11 1-13

Fano variety 1-12: quartic double solid

double cover of $\mathbb{P}^3$ with branch locus a smooth quartic surface

Alternative description:

• hypersurface of degree 4 in $\mathbb{P}(1,1,1,1,2)$

Picard rank

1 (others)

$-\mathrm{K}_X^3$

16

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

10

Mori–Mukai $d$

4

The holomorphic Poisson structures form the following irreducible components of $\mathbb{P}\mathrm{H}^0(X,\wedge^2\mathrm{T}_X)$:

See Loray–Pereira–Touzet.

index

2

del Pezzo of degree 2

$X\overset{2:1}{\to}\mathbb{P}^3$

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

11

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

yes

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

yes

hyperelliptic

no

trigonal

no

This variety is not rational but unirational.

This variety is primitive.

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

• 2-3, 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

19

Bott vanishing

does not hold

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

GRDB

#140

Fanosearch

#1

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

$\mathbb{P}(1^4,2)$

bundle

$\mathcal{O}(4)$

variety

$\mathbb{P}^3 \times \mathbb{P}^{10}$

bundle

$\Lambda(0,1) \oplus \mathcal{O}(0,2)$

See the big table for more information.

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

• the book