Fanography

1-17 2-2

Fano variety 2-1

blowup of 1-11 in an elliptic curve which is the intersection of two divisors from half the anticanonical linear system

Picard rank

2 (others)

$-\mathrm{K}_X^3$

4

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

22

index

1

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

5

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

no

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

no

hyperelliptic

no

trigonal

no

This variety is not known to be unirational.

This variety is the blowup of

• 1-11, in a curve of genus 1

number of moduli

36

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

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

bundle

$\mathcal{O}(6,0) \oplus \mathcal{O}(1,1)$

See the big table for more information.

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

• the book
• 2210.14770