Toric Fano threefolds
The table lists the 18 smooth toric Fano threefolds. The diagrams use the lattice vertices in the corresponding Graded Ring Database records; each row links directly to its record. Drag a diagram to rotate the polytope.
| ID | $\rho$ | $g$ | index | GRDB | polytope | description |
|---|---|---|---|---|---|---|
| 1-17 | 1 | 33 | 4 | #23 | projective space $\mathbb{P}^3$ | |
| 2-36 | 2 | 32 | 1 | #7 | $\mathbb{P}(\mathcal{O}_{\mathbb{P}^2}\oplus\mathcal{O}_{\mathbb{P}^2}(2))$ | |
| 2-35 | 2 | 29 | 2 | #20 | $\mathrm{Bl}_p\mathbb{P}^3$
| |
| 2-33 | 2 | 28 | 1 | #19 | blowup of 1-17 in a line | |
| 2-34 | 2 | 28 | 1 | #22 | $\mathbb{P}^1\times\mathbb{P}^2$ | |
| 3-31 | 3 | 27 | 1 | #11 | blowup of the cone over a smooth quadric in $\mathbb{P}^3$ in the vertex
| |
| 3-29 | 3 | 26 | 1 | #6 | blowup of 2-35 in a line on the exceptional divisor | |
| 3-30 | 3 | 26 | 1 | #12 | blowup of 2-35 in the proper transform of a line containing the center of the blowup
| |
| 3-27 | 3 | 25 | 2 | #21 | $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$ | |
| 3-28 | 3 | 25 | 1 | #17 | $\mathbb{P}^1\times\mathrm{Bl}_p\mathbb{P}^2$ | |
| 3-26 | 3 | 24 | 1 | #16 | blowup of 1-17 in the disjoint union of a point and a line
| |
| 4-12 | 4 | 24 | 1 | #8 | blowup of 2-33 in the disjoint union of two exceptional lines of the blowup | |
| 3-25 | 3 | 23 | 1 | #18 | blowup of 1-17 in the disjoint union of two lines
| |
| 4-11 | 4 | 23 | 1 | #10 | blowup of 3-28 in $\{x\}\times E$, $x\in\mathbb{P}^1$ and $E$ the $(-1)$-curve | |
| 4-10 | 4 | 22 | 1 | #14 | $\mathbb{P}^1\times\mathrm{Bl}_2\mathbb{P}^2$ | |
| 4-9 | 4 | 21 | 1 | #13 | blowup of 3-25 in an exceptional curve of the blowup | |
| 5-2 | 5 | 19 | 1 | #9 | blowup of 3-25 in the disjoint union of two exceptional lines on the same irreducible component | |
| 5-3 | 5 | 19 | 1 | #15 | $\mathbb{P}^1\times\mathrm{Bl}_3\mathbb{P}^2$ |