Fanography

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

Fano threefolds with $\rho=1$

ID $-\mathrm{K}_X^3$ $\mathrm{h}^{1,2}$ index description
1-1 2 52 1

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

1-2 4 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
1-3 6 20 1

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

1-4 8 14 1

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

1-5 10 10 1 Gushel--Mukai 3-fold
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
1-6 12 7 1

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

1-7 14 5 1

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

1-8 16 3 1

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

1-9 18 2 1

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

1-10 22 0 1

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

1-11 8 21 2

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

1-12 16 10 2

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

1-13 24 5 2

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

1-14 32 2 2

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

1-15 40 0 2

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

1-16 54 0 3

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

1-17 64 0 4

projective space $\mathbb{P}^3$

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

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

Fano threefolds with $\rho=3$

ID $-\mathrm{K}_X^3$ $\mathrm{h}^{1,2}$ index description
3-1 12 8 1

double cover of 3-27 with branch locus a divisor of degree $(2,2,2)$

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 $Y\in|\mathcal{L}|$

3-3 18 3 1

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

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

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

3-6 22 1 1

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

3-7 24 1 1

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

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

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 $\mathbb{P}^2$

3-10 26 0 1

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

3-11 28 1 1

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

3-12 28 0 1

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

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 $i=1,2$

3-14 32 1 1

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

3-15 32 0 1

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

3-16 34 0 1

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

3-17 36 0 1

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

3-18 36 0 1

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

3-19 38 0 1

blowup of 1-16 in two non-collinear points

3-20 38 0 1

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

3-21 38 0 1

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

3-22 40 0 1

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

3-23 42 0 1

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

3-24 42 0 1

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

3-25 44 0 1

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

3-26 46 0 1

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

3-27 48 0 2

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

3-28 48 0 1

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

3-29 50 0 1

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

3-30 50 0 1

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

3-31 52 0 1

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

Fano threefolds with $\rho=4$

ID $-\mathrm{K}_X^3$ $\mathrm{h}^{1,2}$ description
4-1 24 1

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

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

4-3 30 0

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

4-4 32 0

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

4-5 32 0

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

4-6 34 0

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

4-7 36 0

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

4-8 38 0

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

4-9 40 0

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

4-10 42 0

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

4-11 44 0

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

4-12 46 0

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

4-13 26 0

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

Fano threefolds with $\rho=5$

ID $-\mathrm{K}_X^3$ $\mathrm{h}^{1,2}$ description
5-1 28 0

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

5-2 36 0

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

5-3 36 0

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

Fano threefolds with $\rho=6$

ID $-\mathrm{K}_X^3$ $\mathrm{h}^{1,2}$ description
6-1 30 0

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

Fano threefolds with $\rho=7$

ID $-\mathrm{K}_X^3$ $\mathrm{h}^{1,2}$ description
7-1 24 0

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

Fano threefolds with $\rho=8$

ID $-\mathrm{K}_X^3$ $\mathrm{h}^{1,2}$ description
8-1 18 0

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

Fano threefolds with $\rho=9$

ID $-\mathrm{K}_X^3$ $\mathrm{h}^{1,2}$ description
9-1 12 0

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

Fano threefolds with $\rho=10$

ID $-\mathrm{K}_X^3$ $\mathrm{h}^{1,2}$ description
10-1 6 0

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