# 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
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