Fano variety 2-8
- double cover of 2-35 with branch locus an anticanonical divisor such that the intersection with the exceptional divisor is smooth
- double cover of 2-35 with branch locus an anticanonical divisor such that the intersection with the exceptional divisor is singular but reduced
- Picard rank
- 2 (others)
- $-\mathrm{K}_X^3$
- 14
- $\mathrm{h}^{1,2}(X)$
- 9
- Mori–Mukai $d$
- -8
0 0
0 2 0
0 9 9 0
0 2 0
0 0
1
0 0
0 18 3
0 0 1 10
0 1 0
0 0
0
Still open.
- index
- 1
- $\dim\mathrm{H}^0(X,\omega_X^\vee)$
- 10
- $-\mathrm{K}_X$ very ample?
- yes
- $-\mathrm{K}_X$ basepoint free?
- yes
- hyperelliptic
- no
- trigonal
- no
This variety is not rational but unirational.
A very general member is not stably rational.
The Weyl group associated to the KKMR decomposition is $\{e\}$; see the Weyl-group table for sources and conventions.
This variety is primitive.
- number of moduli
- 18
- 17
- Bott vanishing
- does not hold
| $\mathrm{Aut}^0(X)$ | $\dim\mathrm{Aut}^0(X)$ | number of moduli |
|---|---|---|
| $0$ | 0 | 18 |
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
- $\mathbb{P}^2 \times \mathbb{P}^3 \times \mathbb{P}^{12}$
- bundle
- $\Lambda(0,0,1) \oplus \mathcal{O}(0,0,2)$
See the big table for more information.
- every member is K‑stable
- every member is K‑polystable
- every member is K‑semistable
For this deformation family, every smooth member has coregularity 0.
The generic and uniform statements are due to Avilov–Loginov–Przyjalkowski; the existence of a smooth member of coregularity zero in every family is due to Zhakupov.
See coregularity for terminology.
For every K-polystable member, the Futaki invariant vanishes identically on the entire Kähler cone.
This is the computation of Sektnan–Tipler.