Fanography

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

Toric Fano threefolds

ID $\rho$ $g$ index description
1-17 1 33 1

projective space $\mathbb{P}^3$

2-36 2 32 2

$\mathbb{P}(\mathcal{O}_{\mathbb{P}^2}\oplus\mathcal{O}_{\mathbb{P}^2}(2))$

2-35 2 29 2

$\mathrm{Bl}_p\mathbb{P}^3$

alternative
$\mathbb{P}(\mathcal{O}_{\mathbb{P}^2}\oplus\mathcal{O}_{\mathbb{P}^2}(1))$
2-33 2 28 2

blowup of 1-17 in a line

2-34 2 28 2

$\mathbb{P}^1\times\mathbb{P}^2$

3-31 3 27 3

blowup of the cone over a smooth quadric in $\mathbb{P}^3$ in the vertex

alternative
$\mathbb{P}(\mathcal{O}\oplus\mathcal{O}(1,1))$ over $\mathbb{P}^1\times\mathbb{P}^1$
3-29 3 26 3

blowup of 2-35 in a line on the exceptional divisor

3-30 3 26 3

blowup of 2-35 in the proper transform of a line containing the center of the blowup

alternative
$\mathbb{P}_{\mathbb{F}_1}(\mathcal{O}\oplus\mathcal{O}(\ell))$ where $\ell^2=1$
3-27 3 25 3

$\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$

3-28 3 25 3

$\mathbb{P}^1\times\mathrm{Bl}_p\mathbb{P}^2$

3-26 3 24 3

blowup of 1-17 in the disjoint union of a point and a line

alternative
blowup of line on a plane which is section of 2-34 mapping to $\mathbb{P}^2$
3-25 3 23 3

blowup of 1-17 in the disjoint union of two lines

alternative
$\mathbb{P}(\mathcal{O}(1,0)\oplus\mathcal{O}(0,1))$ over $\mathbb{P}^1\times\mathbb{P}^1$
4-12 4 24 4

blowup of 2-33 in the disjoint union of two exceptional lines of the blowup

4-11 4 23 4

blowup of 3-28 in $\{x\}\times E$, $x\in\mathbb{P}^1$ and $E$ the $(-1)$-curve

4-10 4 22 4

$\mathbb{P}^1\times\mathrm{Bl}_2\mathbb{P}^2$

4-9 4 21 4

blowup of 3-25 in an exceptional curve of the blowup

5-3 5 19 5

$\mathbb{P}^1\times\mathrm{Bl}_3\mathbb{P}^2$

5-2 5 19 5

blowup of 3-25 in the disjoint union of two exceptional lines on the same irreducible component