Fanography

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

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 3-27 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

alternative
complete intersection of degree $(1,0,2)$ and $(0,1,1)$ in $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^3$
 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 $\pi_i\colon\mathrm{Bl}_1\mathbb{P}^2\times\mathbb{P}^2\to\mathrm{Bl}_1\mathbb{P}^2,\mathbb{P}^2$ 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

alternative
complete intersection of degree $(1,0,1)$, $(0,1,1)$ and $(0,0,2)$ in $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^4$
 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 $\mathbb{P}^2$

alternative
complete intersection of degree $(1,1,0)$ and $(0,1,1)$ in $\mathbb{P}^1\times\mathbb{P}^2\times\mathbb{P}^2$
 * 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 $\mathrm{PGL}_{(2,2)}$
3-26 46 0 1

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

alternative
blowup of line on a plane which is section of 2-34 mapping to $\mathbb{P}^2$
 * 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

alternative
$\mathbb{P}_{\mathbb{F}_1}(\mathcal{O}\oplus\mathcal{O}(\ell))$ where $\ell^2=1$
 * 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 $\mathrm{PSO}_{6;1}$