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
| no | ? | 68 | $0$ | |||||||||
| 1-2 | 4 | 3 | 30 | 1 |
| no | some |
| $0$ | |||||||||
| 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
| generically non-rational | yes |
| $0$ | |||||||||
| 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 |
| |||||||||
| 1-11 | 8 | 21 | 2 | double Veronese cone hypersurface of degree 6 in $\mathbb{P}(1,1,1,2,3)$ | * | no | ? | 34 | $0$ | |||||||||
| 1-12 | 16 | 10 | 2 | quartic double solid double cover of $\mathbb{P}^3$ with branch locus a smooth quartic surface
| * | 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 | quintic del Pezzo threefold 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 | $\mathrm{PGL}_4$ |