Identification
Hodge diamond and polyvector parallelogram
1
0 0
0 7 0
0 0 0 0
0 7 0
0 0
1
0 0
0 7 0
0 0 0 0
0 7 0
0 0
1
1
0 3
0 2 5
0 0 6 15
0 0 0
0 0
0
0 3
0 2 5
0 0 6 15
0 0 0
0 0
0
Anticanonical bundle
- index
- 1
- $\dim\mathrm{H}^0(X,\omega_X^\vee)$
- 15
- $-\mathrm{K}_X$ very ample?
- yes
- $-\mathrm{K}_X$ basepoint free?
- yes
- hyperelliptic
- no
- trigonal
- no
Birational geometry
Deformation theory
- number of moduli
- 2
- Bott vanishing
- does not hold
$\mathrm{Aut}^0(X)$ | $\dim\mathrm{Aut}^0(X)$ | number of moduli |
---|---|---|
$\mathrm{PGL}_2$ | 3 | 2 |
Period sequence
Extremal contractions
$\mathbb{P}^1$-bundle over $\mathrm{Bl}_5\mathbb{P}^2$, for the vector bundle $\mathcal{O}_{\mathrm{Bl}_5\mathbb{P}^2}\oplus\mathcal{O}_{\mathrm{Bl}_5\mathbb{P}^2}$.
Semiorthogonal decompositions
A full exceptional collection can be constructed using Orlov's blowup formula.
Structure of quantum cohomology
Generic semisimplicity of:
- small quantum cohomology, proved by Ciolli in 2005, see [MR2168069] , using the description of quantum cohomology of a $\mathbb{P}^1$-bundle
Zero section description
Fano 3-folds from homogeneous vector bundles over Grassmannians gives the following description(s):
- variety
- $\mathbb{P}^4 \times \mathbb{P}^1$
- bundle
- $\mathcal{O}(2,0)^{\oplus 2}$
See the big table for more information.
K-stability
- none are K-stable
- every member is K‑polystable
- every member is K‑semistable