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 blowups blowdowns rational unirational moduli $\mathrm{Aut}^0$
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

1-11 no ? 26 $0$
2-2 6 20 1

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

no yes 33 $0$
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

1-12 no yes 23 $0$
2-4 10 10 1

blowup of 1-17 in the intersection of two cubics

alternative
$(1,3)$-divisor on $\mathbb{P}^1\times\mathbb{P}^3$
1-17 yes yes 21 $0$
2-5 12 6 1

blowup of 1-13 in a plane cubic

1-13 no yes 16 $0$
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
no yes
  1. 19
  2. 18
$0$
2-7 14 5 1

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

1-16 yes yes 14 $0$
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
no yes
  1. 18
  2. 17
$0$
2-9 16 5 1

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

alternative
blowup of 1-17 in a curve of degree 7 and genus 5, which is intersection of 3 cubics
1-17 yes yes 13 $0$
2-10 16 3 1

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

1-14 yes yes 11 $0$
2-11 18 5 1

blowup of 1-13 in a line

1-13 no yes 12 $0$
2-12 20 3 1

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

alternative
blowup of 1-17 in a curve of degree 6 and genus 3 which is an intersection of 4 cubics
1-17 yes yes 9 $0$
2-13 20 2 1

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

1-16 yes yes 8 $0$
2-14 20 1 1

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

1-15 yes yes 7 $0$
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
1-17 yes yes
  1. 9
  2. 8
$0$
2-16 22 2 1

blowup of 1-14 in a conic

1-14 yes yes 7 $0$
2-17 24 1 1

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

1-16, 1-17 yes yes 5 $0$
2-18 24 2 1

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

* yes yes 6 $0$
2-19 26 2 1

blowup of 1-14 in a line

1-14, 1-17 yes yes 5 $0$
2-20 26 0 1

blowup of 1-15 in a twisted cubic

1-15 yes yes 3
$\mathrm{Aut}^0(X)$ moduli
$\mathbb{G}_{\mathrm{m}}$ 0
2-21 28 0 1

blowup of 1-16 in a twisted quartic

1-16 yes yes 2
$\mathrm{Aut}^0(X)$ moduli
$\mathrm{PGL}_2$ 0
$\mathbb{G}_{\mathrm{a}}$ 0
$\mathbb{G}_{\mathrm{m}}$ 1
2-22 30 0 1

blowup of 1-15 in a conic

1-15, 1-17 yes yes 1
$\mathrm{Aut}^0(X)$ moduli
$\mathbb{G}_{\mathrm{m}}$ 0
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
1-16 yes yes
  1. 2
  2. 1
$0$
2-24 30 0 1

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

* yes yes 1
$\mathrm{Aut}^0(X)$ moduli
$\mathbb{G}_{\mathrm{m}}^2$ 0
$\mathbb{G}_{\mathrm{m}}$ 0
2-25 32 1 1

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

alternative
$(1,2)$-divisor on $\mathbb{P}^1\times\mathbb{P}^3$
* 1-17 yes yes 1 $0$
2-26 34 0 1

blowup of 1-15 in a line

1-15, 1-16 yes yes 0
$\mathrm{Aut}^0(X)$ moduli
$\mathrm{B}$ 0
$\mathbb{G}_{\mathrm{m}}$ 0
2-27 38 0 1

blowup of 1-17 in a twisted cubic

* 1-17 yes yes 0 $\mathrm{PGL}_2$
2-28 40 1 1

blowup of 1-17 in a plane cubic

1-17 yes yes 1 $\mathbb{G}_{\mathrm{a}}^3\rtimes\mathbb{G}_{\mathrm{m}}$
2-29 40 0 1

blowup of 1-16 in a conic

* 1-16 yes yes 0 $\mathbb{G}_{\mathrm{m}}\times\mathrm{PGL}_2$
2-30 46 0 1

blowup of 1-17 in a conic

* 1-17 yes yes 0 $\mathrm{PSO}_{5;1}$
2-31 46 0 1

blowup of 1-16 in a line

* 1-16 yes yes 0 $\mathrm{PSO}_{5;2}$
2-32 48 0 2

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

alternative
$\mathbb{P}(\mathrm{T}_{\mathbb{P}^2})$
the complete flag variety for $\mathbb{P}^2$
* yes yes 0 $\mathrm{PGL}_3$
2-33 54 0 1

blowup of 1-17 in a line

* 1-17 yes yes 0 $\mathrm{PGL}_{4;2}$
2-34 54 0 1

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

* yes yes 0 $\mathrm{PGL}_2\times\mathrm{PGL}_3$
2-35 56 0 2

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

alternative
$\mathbb{P}(\mathcal{O}_{\mathbb{P}^2}\oplus\mathcal{O}_{\mathbb{P}^2}(1))$
* yes yes 0 $\mathrm{PGL}_{4;1}$
2-36 62 0 1

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

* yes yes 0 $\mathrm{Aut}(\mathbb{P}(1,1,1,2))$