Fanography

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

Bott vanishing holds for a smooth projective variety $X$ if $\mathrm{H}^j(X,\Omega_X^i\otimes\mathcal{L})=0$ for all $j\geq 1$, $i\geq 0$ and $\mathcal{L}\in\operatorname{Pic}(X)$ ample.

The Fano 3-folds for which Bott vanishing holds (resp. fails) were classified by Totaro in Bott vanishing for Fano 3-folds. Earlier he showed that del Pezzo surfaces of degree at least 5 satisfy Bott vanishing.

Fano threefolds satisfying Bott vanishing

ID index toric description
1-17 4 true

projective space $\mathbb{P}^3$

2-26 1 false

blowup of 1-15 in a line

2-30 1 false

blowup of 1-17 in a conic

2-33 1 true

blowup of 1-17 in a line

2-34 1 true

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

2-35 2 true

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

2-36 1 true

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

3-15 1 false

blowup of 1-16 in the disjoint union of a line and a conic

3-16 1 false

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

3-18 1 false

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

3-19 1 false

blowup of 1-16 in two non-collinear points

3-20 1 false

blowup of 1-16 in the disjoint union of two lines

3-21 1 false

blowup of 2-34 in a curve of degree $(2,1)$

3-22 1 false

blowup of 2-34 in a conic on $\{x\}\times\mathbb{P}^2$, $x\in\mathbb{P}^1$

3-23 1 false

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

3-24 1 false

the fiber product of 2-32 with $\mathrm{Bl}_p\mathbb{P}^2$ over $\mathbb{P}^2$

3-25 1 true

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

3-26 1 true

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

3-27 2 true

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

3-28 1 true

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

3-29 1 true

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

3-30 1 true

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

3-31 1 true

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

4-3 1 false

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

4-4 1 false

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

4-5 1 false

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

4-6 1 false

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

4-7 1 false

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

4-8 1 false

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

4-9 1 true

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

4-10 1 true

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

4-11 1 true

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

4-12 1 true

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

5-1 1 false

blowup of 2-29 in the disjoint union of three exceptional lines of the blowup

5-2 1 true

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

5-3 1 true

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

6-1 1 false

$\mathbb{P}^1\times\mathrm{Bl}_4\mathbb{P}^2$