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}$indexdescriptionblowupsrationalunirationalmoduli$\mathrm{Aut}^0$
1-122521

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$0$
1-243301
  1. hypersurface of degree 4 in $\mathbb{P}^4$
  2. double cover of 1-16 with branch locus a divisor of degree 8
nosome
  1. 45
  2. 44
$0$
1-364201

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

noyes34$0$
1-485141

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

noyes27$0$
1-5106101Gushel–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-rationalyes
  1. 22
  2. 19
$0$
1-612771

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

yesyes18$0$
1-714851

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

noyes15$0$
1-816931

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

yesyes12$0$
1-9181021

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

yesyes10$0$
1-10221201

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

yesyes6
$\mathrm{Aut}^0(X)$moduli
$\mathrm{PGL}_2$0
$\mathbb{G}_{\mathrm{a}}$0
$\mathbb{G}_{\mathrm{m}}$1
1-118212double Veronese cone

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

*no?34$0$
1-1216102quartic double solid

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

alternative
hypersurface of degree 4 in $\mathbb{P}(1,1,1,1,2)$
*noyes19$0$
1-132452

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

*noyes10$0$
1-143222

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

*yesyes3$0$
1-154002quintic del Pezzo threefold

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

*yesyes0$\mathrm{PGL}_2$
1-165403

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

*yesyes0$\mathrm{PSO}_5$
1-176404

projective space $\mathbb{P}^3$

*yesyes0$\mathrm{PGL}_4$