# Fanography

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

## Fano threefolds with $\rho=2$

ID $-\mathrm{K}_X^3$ $\mathrm{h}^{1,2}$ index description
2-1 4 22 1

blowup of 1-11 in an elliptic curve which is the intersection of two divisors from half the anticanonical linear system

2-2 6 20 1

double cover of 2-34 with branch locus a $(2,4)$-divisor

2-3 8 11 1

blowup of 1-12 in an elliptic curve which is the intersection of two divisors from half the anticanonical linear system

2-4 10 10 1

blowup of 1-17 in the intersection of two cubics

2-5 12 6 1

blowup of 1-13 in a plane cubic

2-6 12 9 1 Verra 3-fold
1. $(2,2)$-divisor on $\mathbb{P}^2\times\mathbb{P}^2$
2. double cover of 2-32 with branch locus an anticanonical divisor
2-7 14 5 1

blowup of 1-16 in the intersection of two divisors from $|\mathcal{O}_{X_2}(2)|$

2-8 14 9 1
1. double cover of 2-35 with branch locus an anticanonical divisor such that the intersection with the exceptional divisor is smooth
2. double cover of 2-35 with branch locus an anticanonical divisor such that the intersection with the exceptional divisor is singular but reduced
2-9 16 5 1

complete intersection of degree $(1,1)$ and $(2,1)$ in $\mathbb{P}^3\times\mathbb{P}^2$

2-10 16 3 1

blowup of 1-14 in an elliptic curve which is intersection of 2 hyperplanes

2-11 18 5 1

blowup of 1-13 in a line

2-12 20 3 1

intersection of 3 $(1,1)$-divisors in $\mathbb{P}^3\times\mathbb{P}^3$

2-13 20 2 1

blowup of 1-16 in a curve of degree 6 and genus 2

2-14 20 1 1

blowup of 1-15 in an elliptic curve which is an intersection of 2 hyperplanes

2-15 22 4 1
1. blowup of 1-17 in the intersection of a quadric and a cubic where the quadric is smooth
2. blowup of 1-17 in the intersection of a quadric and a cubic where the quadric is singular but reduced
2-16 22 2 1

blowup of 1-14 in a conic

2-17 24 1 1

blowup of 1-16 in an elliptic curve of degree 5

2-18 24 2 1

double cover of 2-34 with branch locus a divisor of degree $(2,2)$

2-19 26 2 1

blowup of 1-14 in a line

2-20 26 0 1

blowup of 1-15 in a twisted cubic

2-21 28 0 1

blowup of 1-16 in a twisted quartic

2-22 30 0 1

blowup of 1-15 in a conic

2-23 30 1 1
1. blowup of 1-16 in an intersection of $A\in|\mathcal{O}_Q(1)|$ and $B\in|\mathcal{O}_Q(2)|$ such that $A$ is smooth
2. blowup of 1-16 in an intersection of $A\in|\mathcal{O}_Q(1)|$ and $B\in|\mathcal{O}_Q(2)|$ such that $A$ is singular
2-24 30 0 1

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

2-25 32 1 1

blowup of 1-17 in an elliptic curve which is an intersection of 2 quadrics

2-26 34 0 1

blowup of 1-15 in a line

2-27 38 0 1

blowup of 1-17 in a twisted cubic

2-28 40 1 1

blowup of 1-17 in a plane cubic

2-29 40 0 1

blowup of 1-16 in a conic

2-30 46 0 1

blowup of 1-17 in a conic

2-31 46 0 1

blowup of 1-16 in a line

2-32 48 0 2

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

2-33 54 0 1

blowup of 1-17 in a line

2-34 54 0 1

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

2-35 56 0 2

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

2-36 62 0 1

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