Fanography

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

Identification

Fano variety 3-9

blowup of the cone over the Veronese of $\mathbb{P}^2$ in $\mathbb{P}^5$ with center the disjoint union of the vertex and a quartic curve on $\mathbb{P}^2$

rank
3 (others)
$-\mathrm{K}_X^3$
26
$\mathrm{h}^{1,2}(X)$
3
Hodge diamond
1
0 0
0 3 0
0 3 3 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

  • 2-36, in a curve of genus 3
Deformation theory
number of moduli
6

$\mathrm{Aut}^0(X)$ $\dim\mathrm{Aut}^0(X)$ number of moduli
$\mathbb{G}_{\mathrm{m}}$ 1 6