Fanography

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

description degree description anticanonical model number of exceptional lines moduli $\dim\mathrm{Aut}$ automorphism group
$\mathbb{P}^2$ 9 projective plane

Segre embedding of degree $3$ for $\mathbb{P}^2$ in $\mathbb{P}^9$

0 0 8 $\mathrm{PGL}_3$
$\mathbb{P}^1\times\mathbb{P}^1$ 8 quadric surface

Segre embedding of degree $2$ for $\mathbb{P}^3$ in $\mathbb{P}^8$ restricted to $\mathbb{P}^1\times\mathbb{P}^1$

0 0 6 $(\mathrm{PGL}_2\times\mathrm{PGL}_2)\rtimes\mathbb{Z}/2\mathbb{Z}$
$\mathrm{Bl}_1\mathbb{P}^2$ 8
alternative descriptions
divisor of degree $(1,1)$ in $\mathbb{P}^1\times\mathbb{P}^2$

1 0 6 $\mathbb{A}^2\rtimes\mathrm{GL}_2$
$\mathrm{Bl}_2\mathbb{P}^2$ 7
alternative descriptions
complete intersection of $(1,0,1)$- and $(0,1,1)$-divisor in $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^2$

3 0 4 $\left\{ \Bigl( \begin{smallmatrix} 1 & 0 & * \\ 0 & * & * \\ 0 & 0 & * \end{smallmatrix} \Bigr) \right\}\rtimes\mathrm{Sym}_2$
$\mathrm{Bl}_3\mathbb{P}^2$ 6
alternative descriptions
divisor of degree $(1,1,1)$ in $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$
complete intersection of two $(1,1)$-divisors in $\mathbb{P}^2\times\mathbb{P}^2$

6 0 2 $(\mathbb{G}_{\mathrm{m}}^2\rtimes\mathrm{Sym}_3)\times\mathrm{Sym}_2$
$\mathrm{Bl}_4\mathbb{P}^2$ 5
alternative descriptions
divisor of degree $(1,2)$ in $\mathbb{P}^1\times\mathbb{P}^2$
section of $\mathrm{Gr}(2,5)$ in $\mathbb{P}^9$ by a codimension 4 linear subspace

10 0 0 $\mathrm{Sym}_5$
$\mathrm{Bl}_5\mathbb{P}^2$ 4 Segre quartic surface

intersection of 2 quadrics in $\mathbb{P}^4$

16 2 0 finite
$\mathrm{Bl}_6\mathbb{P}^2$ 3 cubic surface
alternative descriptions
triple cover of $\mathbb{P}^2$ branched along a sextic with six cusps lying on a smooth conic

cubic surface in $\mathbb{P}^3$

27 4 0 finite
$\mathrm{Bl}_7\mathbb{P}^2$ 2 del Pezzo double plane
alternative descriptions
quartic surface in $\mathbb{P}(1,1,1,2)$
divisor of degree $(2,2)$ in $\mathbb{P}^1\times\mathbb{P}^2$

double cover of $\mathbb{P}^2$ branched along a quartic curve

56 6 0 finite
$\mathrm{Bl}_8\mathbb{P}^2$ 1
alternative descriptions
sextic surface in $\mathbb{P}(1,1,2,3)$

$\mathbb{P}^1$

240 8 0 finite

General position

The great debate

Regarding the spelling of del Pezzo (some authors write Del Pezzo), Miles Reid writes the following in §0.7 of Nonnormal del Pezzo varieties:

Rend, del circolo matematico di Palermo 1 (1887), p. 382 records the admission to the circle of dottore Pasquale del Pezzo, marchese di Campodisola. It would be interesting to know why Corrado Segre writing in the same volume (p. 218, 220, 221), along with every subsequent Italian writer, spells the Marquis' name incorrectly with a capital D.