Fanography

A tool to visually study the geography of Fano 3-folds.

Identification

Fano variety 1-5: Gushel--Mukai 3-fold
  1. section of Plücker embedding of $\mathrm{Gr}(2,5)$ by codimension 2 subspace and a quadric
  2. double cover of 1-15 with branch locus an anticanonical divisor
rank
1 (others)
$-\mathrm{K}_X^3$
10
$\mathrm{h}^{1,2}(X)$
10
Hodge diamond
1
0 0
0 1 0
0 10 10 0
0 1 0
0 0
1
Anticanonical bundle
index
1
$\dim\mathrm{H}^0(X,\omega_X^\vee)$
8
$-\mathrm{K}_X$ very ample?
yes
$-\mathrm{K}_X$ basepoint free?
yes
hyperelliptic
no
trigonal
no
Birational geometry

This variety is known to be generically non-rational.


This variety is primitive.


This variety is fibre-like, i.e. it can appear as the fibre of a Mori fibre space.

Deformation theory
number of moduli
  1. 22
  2. 19

$\mathrm{Aut}^0(X)$ $\dim\mathrm{Aut}^0(X)$ number of moduli
$0$ 0 22