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

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 |