Fanography

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

Fano threefolds with $\rho=1$

ID $-\mathrm{K}_X^3$ $g$ $\mathrm{h}^{1,2}$ index description blowups rational unirational moduli $\mathrm{Aut}^0$
1-1 2 2 52 1

double cover of $\mathbb{P}^3$ with branch locus a divisor of degree 6

alternative
hypersurface of degree 6 in $\mathbb{P}(1,1,1,1,3)$
no ? 68 $0$
1-2 4 3 30 1
  1. hypersurface of degree 4 in $\mathbb{P}^4$
  2. double cover of 1-16 with branch locus a divisor of degree 8
no some
  1. 45
  2. 44
$0$
1-3 6 4 20 1

complete intersection of quadric and cubic in $\mathbb{P}^5$

no yes 34 $0$
1-4 8 5 14 1

complete intersection of 3 quadrics in $\mathbb{P}^6$

no yes 27 $0$
1-5 10 6 10 1
  1. section of Plücker embedding of $\mathrm{Gr}(2,5)$ by codimension 2 subspace and a quadric
  2. double cover of 1-15 with branch locus an anticanonical divisor
generically non-rational no
  1. 22
  2. 19
$0$
1-6 12 7 7 1

section of half-spinor embedding of a connected component of $\mathrm{OGr}_+(5,10)$ by codimension 7 subspace

yes yes 18 $0$
1-7 14 8 5 1

section of Plücker embedding of $\mathrm{Gr}(2,6)$ by codimension 5 subspace

no yes 15 $0$
1-8 16 9 3 1

section of Plücker embedding of $\mathrm{SGr}(3,6)$ by codimension 3 subspace

yes yes 12 $0$
1-9 18 10 2 1

section of the adjoint $\mathrm{G}_2$-Grassmannian $\mathrm{G}_2\mathrm{Gr}(2,7)$ by codimension 2 subspace

yes yes 10 $0$
1-10 22 12 0 1

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

yes yes 6
$\mathrm{Aut}^0(X)$ moduli
$\mathrm{PGL}_2$ 0
$\mathbb{G}_{\mathrm{a}}$ 0
$\mathbb{G}_{\mathrm{m}}$ 1
1-11 8 21 2

hypersurface of degree 6 in $\mathbb{P}(1,1,1,2,3)$

* no ? 34 $0$
1-12 16 10 2

double cover of $\mathbb{P}^3$ with branch locus a smooth quartic surface

alternative
hypersurface of degree 4 in $\mathbb{P}(1,1,1,1,2)$
* no yes 19 $0$
1-13 24 5 2

hypersurface of degree 3 in $\mathbb{P}^4$

* no yes 10 $0$
1-14 32 2 2

complete intersection of 2 quadrics in $\mathbb{P}^5$

* yes yes 3 $0$
1-15 40 0 2

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

* yes yes 0 $\mathrm{PGL}_2$
1-16 54 0 3

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

* yes yes 0 $\mathrm{PSO}_5$
1-17 64 0 4

projective space $\mathbb{P}^3$

* yes yes 0 $\mathrm{PGL}_4$

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

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

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

Fano threefolds with $\rho=3$

ID $-\mathrm{K}_X^3$ $\mathrm{h}^{1,2}$ index description blowups blowdowns rational unirational moduli $\mathrm{Aut}^0$
3-1 12 8 1

double cover of $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$ with branch locus a divisor of degree $(2,2,2)$

no yes 17 $0$
3-2 14 3 1

divisor from $|\mathcal{L}^{\otimes 2}\otimes\mathcal{O}(2,3)|$ on the $\mathbb{P}^2$-bundle $\mathbb{P}(\mathcal{O}\oplus\mathcal{O}(-1,-1)^{\oplus 2})$ over $\mathbb{P}^1\times\mathbb{P}^1$ such that $X\cap Y$ is irreducible, and $\mathcal{L}$ is the tautological bundle, and $Y\in|\mathcal{L}|$

yes yes 11 $0$
3-3 18 3 1

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

2-34 yes yes 9 $0$
3-4 18 2 1

blowup of 2-18 in a smooth fiber of the composition of the projection to $\mathbb{P}^1\times\mathbb{P}^2$ with the projection to $\mathbb{P}^2$ of the double cover with the projection

2-18 yes yes 8 $0$
3-5 20 0 1

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

2-34 yes yes 5
$\mathrm{Aut}^0(X)$ moduli
$\mathbb{G}_{\mathrm{m}}$ 0
3-6 22 1 1

blowup of 1-17 in the disjoint union of a line and an elliptic curve of degree 4

2-25, 2-33 yes yes 5 $0$
3-7 24 1 1

blowup of 2-32 in an elliptic curve which is the intersection of two divisors from $|-\frac{1}{2}\mathrm{K}_W|$

2-32, 2-34 yes yes 4 $0$
3-8 24 0 1

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

2-24, 2-34 yes yes 3
$\mathrm{Aut}^0(X)$ moduli
$\mathbb{G}_{\mathrm{m}}$ 0
3-9 26 3 1

blowup of the cone over the Veronese of $\mathbb{P}^2$ in $\mathbb{P}^5$ with center the disjoint union of the vertex and a quartic curve on $\mathbb{P}^2$

2-36 yes yes 6 $\mathbb{G}_{\mathrm{m}}$
3-10 26 0 1

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

2-29 yes yes 2
$\mathrm{Aut}^0(X)$ moduli
$\mathbb{G}_{\mathrm{m}}^2$ 0
$\mathbb{G}_{\mathrm{m}}$ 1
3-11 28 1 1

blowup of 2-35 in an elliptic curve which is the intersection of two divisors from $|-\frac{1}{2}\mathrm{K}_{V_7}|$

2-25, 2-34, 2-35 yes yes 2 $0$
3-12 28 0 1

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

2-27, 2-33, 2-34 yes yes 1
$\mathrm{Aut}^0(X)$ moduli
$\mathbb{G}_{\mathrm{m}}$ 0
3-13 30 0 1

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 $i=1,2$

2-32 yes yes 1
$\mathrm{Aut}^0(X)$ moduli
$\mathrm{PGL}_2$ 0
$\mathbb{G}_{\mathrm{a}}$ 0
$\mathbb{G}_{\mathrm{m}}$ 1
3-14 32 1 1

blowup of 1-17 in the disjoint union of a plane cubic curve and a point outside the plane

2-35, 2-36 yes yes 1 $\mathbb{G}_{\mathrm{m}}$
3-15 32 0 1

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

2-29, 2-31, 2-34 yes yes 0 $\mathbb{G}_{\mathrm{m}}$
3-16 34 0 1

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

2-27, 2-32, 2-35 yes yes 0 $\mathrm{B}$
3-17 36 0 1

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

* 2-34 yes yes 0 $\mathrm{PGL}_2$
3-18 36 0 1

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

* 2-29, 2-30, 2-33 yes yes 0 $\mathrm{B}\times\mathbb{G}_{\mathrm{m}}$
3-19 38 0 1

blowup of 1-16 in two non-collinear points

* 2-35 yes yes 0 $\mathbb{G}_{\mathrm{m}}\times\mathrm{PGL}_2$
3-20 38 0 1

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

2-31, 2-32 yes yes 0 $\mathbb{G}_{\mathrm{m}}\times\mathrm{PGL}_2$
3-21 38 0 1

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

* 2-34 yes yes 0 $\mathbb{G}_{\mathrm{a}}^2\rtimes\mathbb{G}_{\mathrm{m}}^2$
3-22 40 0 1

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

2-34, 2-36 yes yes 0 $\mathrm{B}\times\mathrm{PGL}_2$
3-23 42 0 1

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

2-30, 2-31, 2-35 yes yes 0 $\mathbb{G}_{\mathrm{a}}^3\rtimes(\mathrm{B}\times\mathbb{G}_{\mathrm{m}})$
3-24 42 0 1

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

* 2-32, 2-34 yes yes 0 $\mathrm{PGL}_{3;1}$
3-25 44 0 1

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

alternative
$\mathbb{P}(\mathcal{O}(1,0)\oplus\mathcal{O}(0,1))$ over $\mathbb{P}^1\times\mathbb{P}^1$
* 2-33 yes yes 0 $\mathrm{PGL}_{3;1}$
3-26 46 0 1

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

* 2-34, 2-35 yes yes 0 $\mathbb{G}_{\mathrm{a}}^3\rtimes(\mathrm{GL}_2\times\mathbb{G}_{\mathrm{m}})$
3-27 48 0 2

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

* yes yes 0 $\mathrm{PGL}_2^3$
3-28 48 0 1

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

* 2-34 yes yes 0 $\mathrm{PGL}_2\times\mathrm{PGL}_{3;1}$
3-29 50 0 1

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

2-35 yes yes 0 $\mathrm{PGL}_{4;3,1}$
3-30 50 0 1

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

* 2-33, 2-35 yes yes 0 $\mathrm{PGL}_{4;2,1}$
3-31 52 0 1

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

alternative
$\mathbb{P}(\mathcal{O}\oplus\mathcal{O}(1,1))$ over $\mathbb{P}^1\times\mathbb{P}^1$
* yes yes 0 $\mathrm{PSO}_{6;1}$

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

Fano threefolds with $\rho=5$

ID $-\mathrm{K}_X^3$ $\mathrm{h}^{1,2}$ description blowups blowdowns rational unirational moduli $\mathrm{Aut}^0$
5-1 28 0

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

4-4, 4-12 yes yes 0 $\mathbb{G}_{\mathrm{m}}$
5-2 36 0

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

4-9, 4-11, 4-12 yes yes 0 $\mathbb{G}_{\mathrm{m}}\times\mathrm{GL}_2$
5-3 36 0

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

* 4-10 yes yes 0 $\mathrm{PGL}_2\times\mathbb{G}_{\mathrm{m}}^2$

Fano threefolds with $\rho=6$

ID $-\mathrm{K}_X^3$ $\mathrm{h}^{1,2}$ description blowups blowdowns rational unirational moduli $\mathrm{Aut}^0$
6-1 30 0

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

* 5-3 yes yes 0 $\mathrm{PGL}_2$

Fano threefolds with $\rho=7$

ID $-\mathrm{K}_X^3$ $\mathrm{h}^{1,2}$ description blowups blowdowns rational unirational moduli $\mathrm{Aut}^0$
7-1 24 0

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

* 6-1 yes yes 2 $\mathrm{PGL}_2$

Fano threefolds with $\rho=8$

ID $-\mathrm{K}_X^3$ $\mathrm{h}^{1,2}$ description blowups blowdowns rational unirational moduli $\mathrm{Aut}^0$
8-1 18 0

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

* 7-1 yes yes 4 $\mathrm{PGL}_2$

Fano threefolds with $\rho=9$

ID $-\mathrm{K}_X^3$ $\mathrm{h}^{1,2}$ description blowups blowdowns rational unirational moduli $\mathrm{Aut}^0$
9-1 12 0

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

* 8-1 yes yes 6 $\mathrm{PGL}_2$

Fano threefolds with $\rho=10$

ID $-\mathrm{K}_X^3$ $\mathrm{h}^{1,2}$ description blowdowns rational unirational moduli $\mathrm{Aut}^0$
10-1 6 0

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

9-1 yes yes 8 $\mathrm{PGL}_2$