Identification
Hodge diamond and polyvector parallelogram
1
0 0
0 3 0
0 0 0 0
0 3 0
0 0
1
0 0
0 3 0
0 0 0 0
0 3 0
0 0
1
1
0 0^{*}
0 1^{*} 7
0 0 0 17
0 0 0
0 0
0
^{*} indicates jumping of $\operatorname{Aut}^0$
its dimension takes on the values 0, 1
0 0^{*}
0 1^{*} 7
0 0 0 17
0 0 0
0 0
0
^{*} indicates jumping of $\operatorname{Aut}^0$
its dimension takes on the values 0, 1
Anticanonical bundle
- index
- 1
- $\dim\mathrm{H}^0(X,\omega_X^\vee)$
- 17
- $-\mathrm{K}_X$ very ample?
- yes
- $-\mathrm{K}_X$ basepoint free?
- yes
- hyperelliptic
- no
- trigonal
- no
Birational geometry
Deformation theory
- number of moduli
- 1
- Bott vanishing
- does not hold
$\mathrm{Aut}^0(X)$ | $\dim\mathrm{Aut}^0(X)$ | number of moduli |
---|---|---|
$\mathbb{G}_{\mathrm{m}}$ | 1 | 0 |
$0$ | 0 | 1 |
Period sequence
Extremal contractions
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 a 1 or 2-curve blowup of $\mathbb{P}^3$ or $Q^3$
Zero section description
Fano 3-folds from homogeneous vector bundles over Grassmannians gives the following description(s):
- variety
- $\mathbb{P}^1 \times \mathbb{P}^2 \times \mathbb{P}^3$
- bundle
- $\mathcal{O}(0,1,1) \oplus \mathcal{O}(0,1,1) \oplus \mathcal{O}(1,0,1)$
See the big table for more information.