Identification
Hodge diamond and polyvector parallelogram
1
0 0
0 2 0
0 0 0 0
0 2 0
0 0
1
0 0
0 2 0
0 0 0 0
0 2 0
0 0
1
1
0 0*
0 2* 8
0 0 0 17
0 0 0
0 0
0
* indicates jumping of $\operatorname{Aut}^0$
its dimension takes on the values 0, 1, 3
0 0*
0 2* 8
0 0 0 17
0 0 0
0 0
0
* indicates jumping of $\operatorname{Aut}^0$
its dimension takes on the values 0, 1, 3
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
This variety is rational.
This variety is the blowup of
- 1-16, 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
- 2
- Bott vanishing
- does not hold
| $\mathrm{Aut}^0(X)$ | $\dim\mathrm{Aut}^0(X)$ | number of moduli |
|---|---|---|
| $\mathrm{PGL}_2$ | 3 | 0 |
| $\mathbb{G}_{\mathrm{a}}$ | 1 | 0 |
| $\mathbb{G}_{\mathrm{m}}$ | 1 | 1 |
| $0$ | 0 | 2 |
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
- $\operatorname{Gr}(2,4) \times \mathbb{P}^4$
- bundle
- $\mathcal{U}^{\vee}_{\operatorname{Gr}(2,4)}(0,1)^{\oplus 2} \oplus \mathcal{O}(1,0)$
See the big table for more information.
K-stability
- general member is K‑stable but there exists one that is not
- general member is K‑polystable but there exists one that is not
- general member is K‑semistable
Hilbert schemes of curves
The Hilbert scheme of conics is a quadric surface.
Its Hodge diamond is
1
0 0
0 2 0
0 0
1
0 0
0 2 0
0 0
1