Identification
Hodge diamond and polyvector parallelogram
1
0 0
0 1 0
0 0 0 0
0 1 0
0 0
1
0 0
0 1 0
0 0 0 0
0 1 0
0 0
1
1
0 3
0 0 21
0 0 0 23
0 0 0
0 0
0
0 3
0 0 21
0 0 0 23
0 0 0
0 0
0
Anticanonical bundle
- index
- 2
- del Pezzo of degree 5
- $X\hookrightarrow\mathbb{P}^{6}$
- $\dim\mathrm{H}^0(X,\omega_X^\vee)$
- 23
- $-\mathrm{K}_X$ very ample?
- yes
- $-\mathrm{K}_X$ basepoint free?
- yes
- hyperelliptic
- no
- trigonal
- no
Birational geometry
This variety is rational.
This variety is primitive.
This variety can be blown up (in a curve) to
- 2-14, in a curve of genus 1
- 2-20, in a curve of genus 0
- 2-22, in a curve of genus 0
- 2-26, in a curve of genus 0
This variety is fibre-like, i.e. it can appear as the fibre of a Mori fibre space.
Deformation theory
- number of moduli
- 0
- Bott vanishing
- does not hold
$\mathrm{Aut}^0(X)$ | $\dim\mathrm{Aut}^0(X)$ | number of moduli |
---|---|---|
$\mathrm{PGL}_2$ | 3 | 0 |
Period sequence
Semiorthogonal decompositions
A full exceptional collection was constructed by Orlov in 1991, see [MR1294662] .
Structure of quantum cohomology
Generic semisimplicity of:
- small quantum cohomology, proved by someone in at some point, see [?] , using
Zero section description
Fano 3-folds from homogeneous vector bundles over Grassmannians gives the following description(s):
- variety
- $\operatorname{Gr}(2,5)$
- bundle
- $\mathcal{O}(1)^{\oplus 3}$
See the big table for more information.
Hilbert schemes of curves
The Hilbert scheme of lines is $\mathbb{P}^2$.
Its Hodge diamond is
1
0 0
0 1 0
0 0
1
0 0
0 1 0
0 0
1