Identification
Hodge diamond and polyvector parallelogram
1
0 0
0 1 0
0 5 5 0
0 1 0
0 0
1
0 0
0 1 0
0 5 5 0
0 1 0
0 0
1
1
0 0
0 10 10
0 0 0 15
0 0 0
0 0
0
0 0
0 10 10
0 0 0 15
0 0 0
0 0
0
Anticanonical bundle
- index
- 2
- del Pezzo of degree 3
- $X\hookrightarrow\mathbb{P}^{4}$
- $\dim\mathrm{H}^0(X,\omega_X^\vee)$
- 15
- $-\mathrm{K}_X$ very ample?
- yes
- $-\mathrm{K}_X$ basepoint free?
- yes
- hyperelliptic
- no
- trigonal
- no
Birational geometry
This variety is not rational but unirational.
This variety is primitive.
This variety can be blown up (in a curve) to
This variety is fibre-like, i.e. it can appear as the fibre of a Mori fibre space.
Deformation theory
- number of moduli
- 10
- Bott vanishing
- does not hold
$\mathrm{Aut}^0(X)$ | $\dim\mathrm{Aut}^0(X)$ | number of moduli |
---|---|---|
$0$ | 0 | 10 |
Period sequence
Semiorthogonal decompositions
There exist interesting semiorthogonal decompositions, but this data is not yet added.
Structure of quantum cohomology
By Hertling–Manin–Teleman we have that quantum cohomology cannot be generically semisimple, as $\mathrm{h}^{1,2}\neq 0$.
Zero section description
Fano 3-folds from homogeneous vector bundles over Grassmannians gives the following description(s):
- variety
- $\mathbb{P}^4$
- bundle
- $\mathcal{O}(3)$
See the big table for more information.
K-stability
- every member is K‑stable
- every member is K‑polystable
- every member is K‑semistable
Hilbert schemes of curves
The Hilbert scheme of lines is a Fano surface, a minimal surface of general type.
Its Hodge diamond is
1
5 5
10 25 10
5 5
1
5 5
10 25 10
5 5
1