Fanography

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

Identification

Fano variety 3-10

blowup of 1-16 in the disjoint union of 2 conics

Alternative description:

  • complete intersection of degree $(1,0,1)$, $(0,1,1)$ and $(0,0,2)$ in $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^4$
rank
3 (others)
$-\mathrm{K}_X^3$
26
$\mathrm{h}^{1,2}(X)$
0
Hodge diamond
1
0 0
0 3 0
0 0 0 0
0 3 0
0 0
1
Anticanonical bundle
index
1
$\dim\mathrm{H}^0(X,\omega_X^\vee)$
16
$-\mathrm{K}_X$ 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

  • 2-29, in a curve of genus 0
Deformation theory
number of moduli
2

$\mathrm{Aut}^0(X)$ $\dim\mathrm{Aut}^0(X)$ number of moduli
$\mathbb{G}_{\mathrm{m}}^2$ 2 0
$\mathbb{G}_{\mathrm{m}}$ 1 1
$0$ 0 2