# 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

1 0 6 $\mathbb{G}_{\mathrm{m}}^2\rtimes\mathrm{GL}_2$
$\mathrm{Bl}_2\mathbb{P}^2$ 7

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

6 0 2 $(\mathbb{G}_{\mathrm{m}}^2\rtimes\mathrm{Sym}_3)\times\mathrm{Sym}_2$
$\mathrm{Bl}_4\mathbb{P}^2$ 5

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

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

27 4 0 finite
$\mathrm{Bl}_7\mathbb{P}^2$ 2 del Pezzo double plane

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

56 6 0 finite
$\mathrm{Bl}_8\mathbb{P}^2$ 1

$\mathbb{P}^1$

240 8 0 finite

## General position

• no three are collinear
• no six on the same conic
• no eight on a cubic with a double point at one of them

## 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.