Fanography

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

Fano threefolds with $\rho=1$

ID $-\mathrm{K}_X^3$ $g$ $\mathrm{h}^{1,2}$ index description blowups rational unirational moduli $\mathrm{Aut}^0$
1-1 2 2 52 1

double cover of $\mathbb{P}^3$ with branch locus a divisor of degree 6

alternative
hypersurface of degree 6 in $\mathbb{P}(1,1,1,1,3)$
no ? 68 $0$
1-2 4 3 30 1
  1. hypersurface of degree 4 in $\mathbb{P}^4$
  2. double cover of 1-16 with branch locus a divisor of degree 8
no some
  1. 45
  2. 44
$0$
1-3 6 4 20 1

complete intersection of quadric and cubic in $\mathbb{P}^5$

no yes 34 $0$
1-4 8 5 14 1

complete intersection of 3 quadrics in $\mathbb{P}^6$

no yes 27 $0$
1-5 10 6 10 1 Gushel--Mukai 3-fold
  1. section of Plücker embedding of $\mathrm{Gr}(2,5)$ by codimension 2 subspace and a quadric
  2. double cover of 1-15 with branch locus an anticanonical divisor
generically non-rational no
  1. 22
  2. 19
$0$
1-6 12 7 7 1

section of half-spinor embedding of a connected component of $\mathrm{OGr}_+(5,10)$ by codimension 7 subspace

yes yes 18 $0$
1-7 14 8 5 1

section of Plücker embedding of $\mathrm{Gr}(2,6)$ by codimension 5 subspace

no yes 15 $0$
1-8 16 9 3 1

section of Plücker embedding of $\mathrm{SGr}(3,6)$ by codimension 3 subspace

yes yes 12 $0$
1-9 18 10 2 1

section of the adjoint $\mathrm{G}_2$-Grassmannian $\mathrm{G}_2\mathrm{Gr}(2,7)$ by codimension 2 subspace

yes yes 10 $0$
1-10 22 12 0 1

zero locus of $(\bigwedge^2\mathcal{U}^\vee)^{\oplus 3}$ on $\mathrm{Gr}(3,7)$

yes yes 6
$\mathrm{Aut}^0(X)$ moduli
$\mathrm{PGL}_2$ 0
$\mathbb{G}_{\mathrm{a}}$ 0
$\mathbb{G}_{\mathrm{m}}$ 1
1-11 8 21 2

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

* no ? 34 $0$
1-12 16 10 2

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

alternative
hypersurface of degree 4 in $\mathbb{P}(1,1,1,1,2)$
* no yes 19 $0$
1-13 24 5 2

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

* no yes 10 $0$
1-14 32 2 2

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

* yes yes 3 $0$
1-15 40 0 2

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

* yes yes 0 $\mathrm{PGL}_2$
1-16 54 0 3

hypersurface of degree 2 in $\mathbb{P}^4$

* yes yes 0 $\mathrm{PSO}_5$
1-17 64 0 4

projective space $\mathbb{P}^3$

* yes yes 0 $\mathrm{PGL}_4$