Fanography

2-11 2-13

Fano variety 2-12

intersection of 3 $(1,1)$-divisors in $\mathbb{P}^3\times\mathbb{P}^3$

Alternative description:

• blowup of 1-17 in a curve of degree 6 and genus 3 which is an intersection of 4 cubics

Picard rank

2 (others)

$-\mathrm{K}_X^3$

20

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

3

index

1

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

13

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

yes

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

yes

hyperelliptic

no

trigonal

no

This variety is rational.

This variety is the blowup of

• 1-17, in a curve of genus 3

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

number of moduli

9

Bott vanishing

does not hold

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

GRDB

#118

Fanosearch

#13

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

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

bundle

$\mathcal{O}(1,1)^{\oplus 3}$

See the big table for more information.

The Hilbert scheme of conics is the symmetric square of a genus 3 curve.

Its Hodge diamond is