# Fanography

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

## del Pezzo threefolds

On this page we list del Pezzo varieties, which can exist in arbitrary dimension, specialised to dimension 3.

We say that a pair $(X,H)$ of a smooth projective variety $X$ and an ample divisor $H$ is a del Pezzo variety if $-\mathrm{K}_X=(\dim X-1)H$. So for Fano 3-folds these have (co)index 2 and $H$ is the generator of the Picard group, or $X=\mathbb{P}^3$ and $H$ is twice the generator of the Picard group.

Closely related are del Pezzo surfaces, which have have their own page.

Fano 3-fold $H^{\dim X}$ $\rho$ dimension description in dimension 3 description
1-11 1 1 any

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

hypersurface of degree 6 in $\mathbb{P}(1,\ldots,1,2,3)$
1-12 2 1 any

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

double cover of $\mathbb{P}^n$ with branch locus a smooth quartic hypersurface
1-13 3 1 any

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

cubic hypersurface
1-14 4 1 any

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

1-15 5 1 $\leq 6$

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

linear section of Plücker embedding of $\mathrm{Gr}(2,5)$
3-27 6 3 $3$

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

2-32 6 2 $\leq 4$

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

linear section of Segre embedding of $\mathbb{P}^2\times\mathbb{P}^2$
2-35 7 2 $3$

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