Fanography

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

Fano threefolds with $\rho=4$

ID $-\mathrm{K}_X^3$ $\mathrm{h}^{1,2}$ description blowups blowdowns rational unirational moduli $\mathrm{Aut}^0$
4-1 24 1

divisor on $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$ of degree $(1,1,1,1)$

3-27 yes yes 3 $0$
4-2 28 1

blowup of the cone over a smooth quadric in $\mathbb{P}^3$ in the disjoint union of the vertex and an elliptic curve on the quadric

3-31 yes yes 1 $\mathbb{G}_{\mathrm{m}}$
4-3 30 0

blowup of 3-27 in a curve of degree $(1,1,2)$

3-17, 3-27, 3-28 yes yes 0 $\mathbb{G}_{\mathrm{m}}$
4-4 32 0

blowup of 3-19 in the proper transform of a conic through the points

* 3-18, 3-19, 3-30 yes yes 0 $\mathbb{G}_{\mathrm{m}}^2$
4-5 32 0

blowup of 2-34 in the disjoint union of a curve of degree $(2,1)$ and a curve of degree $(1,0)$

3-21, 3-28, 3-31 yes yes 0 $\mathbb{G}_{\mathrm{m}}^2$
4-6 34 0

blowup of 1-17 in the disjoint union of 3 lines

alternative
blowup of 3-27 in the tridiagonal
3-25, 3-27 yes yes 0 $\mathrm{PGL}_2$
4-7 36 0

blowup of 2-32 in the disjoint union of a curve of degree $(0,1)$ and a curve of degree $(1,0)$

3-24, 3-28 yes yes 0 $\mathrm{GL}_2$
4-8 38 0

blowup of 3-27 in a curve of degree $(0,1,1)$

3-27, 3-31 yes yes 0 $\mathrm{B}\times\mathrm{PGL}_2$
4-9 40 0

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

* 3-25, 3-26, 3-28, 3-30 yes yes 0 $\mathrm{PGL}_{(2,2);1}$
4-10 42 0

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

* 3-27, 3-28 yes yes 0 $\mathrm{PGL}_2\times\mathrm{B}^2$
4-11 44 0

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

* 3-28, 3-31 yes yes 0 $\mathrm{B}\times\mathrm{PGL}_{3;1}$
4-12 46 0

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

* 3-30 yes yes 0 $\mathbb{G}_{\mathrm{a}}^4\rtimes(\mathrm{GL}_2\times\mathbb{G}_{\mathrm{m}})$
4-13 26 0

blowup of 3-27 in a curve of degree $(1,1,3)$

3-27, 3-31 yes yes 1
$\mathrm{Aut}^0(X)$ moduli
$\mathbb{G}_{\mathrm{m}}$ 0