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 $\mathbb{P}^5$ |
complete intersection of two quadrics |
| 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 | $3$ |
$\mathrm{Bl}_p\mathbb{P}^3$
|