# Fanography

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

## Dubrovin's conjecture

explain this

Picard rank $\rho$ 1 2 3 4 5 6 7 8 9 10 total
number of families 17 36 31 13 3 1 1 1 1 1 105
number of families with $\mathrm{h}^{1,2}=0$ 4 14 22 11 3 1 1 1 1 1 59
number of families with generically semisimple small quantum cohomology 4 13 14 4 2 1 1 1 1 1 42
number of families with full exceptional collections constructed 4 14 22 11 3 1 1 1 1 1 59

For now, only generic semisimplicity of small quantum cohomology is in the table. The generic semisimplicity of quantum cohomology for all the known cases is established at the level of small quantum cohomology, which implies it at the big level (but not conversely). Once Fano 3-folds where only the big quantum cohomology is known to be generically semisimple, the table will be updated accordingly.

ID description existence of a full exceptional collection generic semisimplicity of small quantum cohomology
1-10

zero locus of $(\bigwedge^2\mathcal{U}^\vee)^{\oplus 3}$ on $\mathrm{Gr}(3,7)$

Kuznetsov constructed it in 1996 someone proved it in at some point using
1-15

section of Plücker embedding of $\mathrm{Gr}(2,5)$ by codimension 3 subspace

Orlov constructed it in 1991 someone proved it in at some point using
1-16

hypersurface of degree 2 in $\mathbb{P}^4$

Kapranov constructed it in 1986 someone proved it in at some point using
1-17

projective space $\mathbb{P}^3$

Beilinson constructed it in 1978 someone proved it in at some point using
Iritani proved it in 2007 using toric geometry
2-20

blowup of 1-15 in a twisted cubic

constructed using Orlov's blowup formula not known to hold
2-21

blowup of 1-16 in a twisted quartic

constructed using Orlov's blowup formula Ciolli proved it in 2005 using the description of a 1 or 2-curve blowup of $\mathbb{P}^3$ or $Q^3$
2-22

blowup of 1-15 in a conic

constructed using Orlov's blowup formula Ciolli proved it in 2005 using the description of a 1 or 2-curve blowup of $\mathbb{P}^3$ or $Q^3$
2-24

divisor on $\mathbb{P}^2\times\mathbb{P}^2$ of bidegree $(1,2)$

constructed using Orlov's projective bundle formula Ciolli proved it in 2005 using the description of quantum cohomology of a $\mathbb{P}^1$-bundle
2-26

blowup of 1-15 in a line

constructed using Orlov's blowup formula Ciolli proved it in 2005 using the description of a 1 or 2-curve blowup of $\mathbb{P}^3$ or $Q^3$
2-27

blowup of 1-17 in a twisted cubic

constructed using Orlov's blowup formula Ciolli proved it in 2005 using the description of a 1 or 2-curve blowup of $\mathbb{P}^3$ or $Q^3$
Ciolli proved it in 2005 using the description of quantum cohomology of a $\mathbb{P}^1$-bundle
2-29

blowup of 1-16 in a conic

constructed using Orlov's blowup formula Ciolli proved it in 2005 using the description of a 1 or 2-curve blowup of $\mathbb{P}^3$ or $Q^3$
2-30

blowup of 1-17 in a conic

constructed using Orlov's blowup formula Ciolli proved it in 2005 using the description of a 1 or 2-curve blowup of $\mathbb{P}^3$ or $Q^3$
2-31

blowup of 1-16 in a line

constructed using Orlov's blowup formula Ciolli proved it in 2005 using the description of quantum cohomology of a $\mathbb{P}^1$-bundle
2-32

divisor on $\mathbb{P}^2\times\mathbb{P}^2$ of bidegree $(1,1)$

constructed using Orlov's projective bundle formula Ciolli proved it in 2005 using the description of quantum cohomology of a $\mathbb{P}^1$-bundle
2-33

blowup of 1-17 in a line

constructed using Orlov's blowup formula Ciolli proved it in 2005 using the description of a 1 or 2-curve blowup of $\mathbb{P}^3$ or $Q^3$
Iritani proved it in 2007 using toric geometry
2-34

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

constructed using Orlov's projective bundle formula Iritani proved it in 2007 using toric geometry
Ciolli proved it in 2005 using the description of quantum cohomology of a $\mathbb{P}^1$-bundle
2-35

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

constructed using Orlov's projective bundle formula Iritani proved it in 2007 using toric geometry
Ciolli proved it in 2005 using the description of quantum cohomology of a $\mathbb{P}^1$-bundle
2-36

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

constructed using Orlov's projective bundle formula Iritani proved it in 2007 using toric geometry
Ciolli proved it in 2005 using the description of quantum cohomology of a $\mathbb{P}^1$-bundle
3-5

blowup of 2-34 in a curve $C$ of degree $(5,2)$ such that $C\hookrightarrow\mathbb{P}^1\times\mathbb{P}^2\to\mathbb{P}^2$ is an embedding

constructed using Orlov's blowup formula not known to hold
3-8

divisor from the linear system $|(\alpha\circ\pi_1)^*(\mathcal{O}_{\mathbb{P}^2}(1)\otimes\pi_2^*(\mathcal{O}_{\mathbb{P}^2}(2))|$ where $\pi_i\colon\mathrm{Bl}_1\times\mathbb{P}^2$ are the projections, and $\alpha\colon\mathrm{Bl}_1\mathbb{P}^2\to\mathbb{P}^2$ is the blowup

constructed using Orlov's blowup formula not known to hold
3-10

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

constructed using Orlov's blowup formula Ciolli proved it in 2005 using the description of a 1 or 2-curve blowup of $\mathbb{P}^3$ or $Q^3$
3-12

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

constructed using Orlov's blowup formula Ciolli proved it in 2005 using the description of a 1 or 2-curve blowup of $\mathbb{P}^3$ or $Q^3$
3-13

blowup of 2-32 in a curve $C$ of bidegree $(2,2)$ such that the composition $C\hookrightarrow W\hookrightarrow\mathbb{P}^2\times\mathbb{P}^2\overset{p_i}{\to}\mathbb{P}^2$ is an embedding for $i=1,2$

constructed using Orlov's blowup formula not known to hold
3-15

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

constructed using Orlov's blowup formula Ciolli proved it in 2005 using the description of a 1 or 2-curve blowup of $\mathbb{P}^3$ or $Q^3$
3-16

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

constructed using Orlov's blowup formula not known to hold
3-17

divisor on $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^2$ of degree $(1,1,1)$

constructed using Orlov's blowup formula Ciolli proved it in 2005 using the description of quantum cohomology of a $\mathbb{P}^1$-bundle
3-18

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

constructed using Orlov's blowup formula Ciolli proved it in 2005 using the description of a 1 or 2-curve blowup of $\mathbb{P}^3$ or $Q^3$
3-19

blowup of 1-16 in two non-collinear points

constructed using Orlov's blowup formula not known to hold
3-20

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

constructed using Orlov's blowup formula Ciolli proved it in 2005 using the description of a 1 or 2-curve blowup of $\mathbb{P}^3$ or $Q^3$
3-21

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

constructed using Orlov's blowup formula not known to hold
3-22

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

constructed using Orlov's blowup formula not known to hold
3-23

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

constructed using Orlov's blowup formula not known to hold
3-24

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

constructed using Orlov's blowup formula Ciolli proved it in 2005 using the description of quantum cohomology of a $\mathbb{P}^1$-bundle
3-25

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

constructed using Orlov's blowup formula Ciolli proved it in 2005 using the description of a 1 or 2-curve blowup of $\mathbb{P}^3$ or $Q^3$
Iritani proved it in 2007 using toric geometry
Ciolli proved it in 2005 using the description of quantum cohomology of a $\mathbb{P}^1$-bundle
3-26

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

constructed using Orlov's blowup formula Iritani proved it in 2007 using toric geometry
3-27

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

constructed using Orlov's projective bundle formula Iritani proved it in 2007 using toric geometry
Ciolli proved it in 2005 using the description of quantum cohomology of a $\mathbb{P}^1$-bundle
3-28

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

constructed using Orlov's blowup formula Iritani proved it in 2007 using toric geometry
Ciolli proved it in 2005 using the description of quantum cohomology of a $\mathbb{P}^1$-bundle
3-29

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

constructed using Orlov's blowup formula Iritani proved it in 2007 using toric geometry
3-30

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

constructed using Orlov's blowup formula Iritani proved it in 2007 using toric geometry
Ciolli proved it in 2005 using the description of quantum cohomology of a $\mathbb{P}^1$-bundle
3-31

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

constructed using Orlov's projective bundle formula Iritani proved it in 2007 using toric geometry
Ciolli proved it in 2005 using the description of quantum cohomology of a $\mathbb{P}^1$-bundle
4-3

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

constructed using Orlov's blowup formula not known to hold
4-4

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

constructed using Orlov's blowup formula not known to hold
4-5

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

constructed using Orlov's blowup formula not known to hold
4-6

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

constructed using Orlov's blowup formula not known to hold
4-7

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

constructed using Orlov's blowup formula not known to hold
4-8

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

constructed using Orlov's blowup formula not known to hold
4-9

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

constructed using Orlov's blowup formula Iritani proved it in 2007 using toric geometry
4-10

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

constructed using Orlov's blowup formula Iritani proved it in 2007 using toric geometry
Ciolli proved it in 2005 using the description of quantum cohomology of a $\mathbb{P}^1$-bundle
4-11

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

constructed using Orlov's blowup formula Iritani proved it in 2007 using toric geometry
4-12

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

constructed using Orlov's blowup formula Iritani proved it in 2007 using toric geometry
4-13

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

constructed using Orlov's blowup formula not known to hold
5-1

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

constructed using Orlov's blowup formula not known to hold
5-2

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

constructed using Orlov's blowup formula Iritani proved it in 2007 using toric geometry
5-3

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

constructed using Orlov's blowup formula Iritani proved it in 2007 using toric geometry
Ciolli proved it in 2005 using the description of quantum cohomology of a $\mathbb{P}^1$-bundle
6-1

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

constructed using Orlov's blowup formula Ciolli proved it in 2005 using the description of quantum cohomology of a $\mathbb{P}^1$-bundle
7-1

$\mathbb{P}^1\times\mathrm{Bl}_5\mathbb{P}^2$

constructed using Orlov's blowup formula Ciolli proved it in 2005 using the description of quantum cohomology of a $\mathbb{P}^1$-bundle
8-1

$\mathbb{P}^1\times\mathrm{Bl}_6\mathbb{P}^2$

constructed using Orlov's blowup formula Ciolli proved it in 2005 using the description of quantum cohomology of a $\mathbb{P}^1$-bundle
9-1

$\mathbb{P}^1\times\mathrm{Bl}_7\mathbb{P}^2$

constructed using Orlov's blowup formula Ciolli proved it in 2005 using the description of quantum cohomology of a $\mathbb{P}^1$-bundle
10-1

$\mathbb{P}^1\times\mathrm{Bl}_8\mathbb{P}^2$

constructed using Orlov's blowup formula Ciolli proved it in 2005 using the description of quantum cohomology of a $\mathbb{P}^1$-bundle