Identification

Hodge diamond

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 1

0 1 9

0 0 0 18

0 0 0

0 0

0

0 1

0 1 9

0 0 0 18

0 0 0

0 0

0

Anticanonical bundle

- index
- 1
- $\dim\mathrm{H}^0(X,\omega_X^\vee)$
- 18
- $-\mathrm{K}_X$ 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

- 2-32, 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
- 1

$\mathrm{Aut}^0(X)$ | $\dim\mathrm{Aut}^0(X)$ | number of moduli |
---|---|---|

$\mathrm{PGL}_2$ | 3 | 0 |

$\mathbb{G}_{\mathrm{a}}$ | 1 | 0 |

$\mathbb{G}_{\mathrm{m}}$ | 1 | 1 |

Period sequence

Extremal contractions

Semiorthogonal decompositions

A full exceptional collection can be constructed using Orlov's blowup formula.

Structure of quantum cohomology

Zero section description

Fano 3-folds from homogeneous vector bundles over Grassmannians gives the following description(s):

- variety
- $(\mathbb{P}^2)^3$
- bundle
- $\mathcal{O}(1,1,0) \oplus \mathcal{O}(1,0,1) \oplus \mathcal{O}(0,1,1)$

See the big table for more information.

Hilbert schemes of curves

The **Hilbert scheme of conics** is the disjoint union of 3 projective planes.

Its Hodge diamond is

1

0 0

0 1 0

0 0

1

0 0

0 1 0

0 0

1