Fano threefolds with $\rho=4$
| ID | $-\mathrm{K}_X^3$ | $\mathrm{h}^{1,2}$ | description | blowups | blowdowns | rational | unirational | moduli | $\mathrm{Aut}^0$ | ||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 4-1 | 24 | 1 | divisor on $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$ of degree $(1,1,1,1)$ | 3-27 | yes | yes | 3 | $0$ | |||||
| 4-2 | 28 | 1 | blowup of the cone over a smooth quadric in $\mathbb{P}^3$ in the disjoint union of the vertex and an elliptic curve on the quadric | 3-31 | yes | yes | 2 | $\mathbb{G}_{\mathrm{m}}$ | |||||
| 4-3 | 30 | 0 | blowup of 3-27 in a curve of degree $(1,1,2)$ | 3-17, 3-27, 3-28 | yes | yes | 0 | $\mathbb{G}_{\mathrm{m}}$ | |||||
| 4-4 | 32 | 0 | blowup of 3-19 in the proper transform of a conic through the points | * | 3-18, 3-19, 3-30 | yes | yes | 0 | $\mathbb{G}_{\mathrm{m}}^2$ | ||||
| 4-5 | 32 | 0 | blowup of 2-34 in the disjoint union of a curve of degree $(2,1)$ and a curve of degree $(1,0)$ | 3-21, 3-28, 3-31 | yes | yes | 0 | $\mathbb{G}_{\mathrm{m}}^2$ | |||||
| 4-6 | 34 | 0 | blowup of 1-17 in the disjoint union of 3 lines
| 3-25, 3-27 | yes | yes | 0 | $\mathrm{PGL}_2$ | |||||
| 4-7 | 36 | 0 | blowup of 2-32 in the disjoint union of a curve of degree $(0,1)$ and a curve of degree $(1,0)$ | 3-24, 3-28 | yes | yes | 0 | $\mathrm{GL}_2$ | |||||
| 4-8 | 38 | 0 | blowup of 3-27 in a curve of degree $(0,1,1)$ | 3-27, 3-31 | yes | yes | 0 | $\mathrm{B}\times\mathrm{PGL}_2$ | |||||
| 4-9 | 40 | 0 | blowup of 3-25 in an exceptional curve of the blowup | * | 3-25, 3-26, 3-28, 3-30 | yes | yes | 0 | $\mathrm{PGL}_{(2,2);1}$ | ||||
| 4-10 | 42 | 0 | $\mathbb{P}^1\times\mathrm{Bl}_2\mathbb{P}^2$ | * | 3-27, 3-28 | yes | yes | 0 | $\mathrm{PGL}_2\times\mathrm{B}^2$ | ||||
| 4-11 | 44 | 0 | blowup of 3-28 in $\{x\}\times E$, $x\in\mathbb{P}^1$ and $E$ the $(-1)$-curve | * | 3-28, 3-31 | yes | yes | 0 | $\mathrm{B}\times\mathrm{PGL}_{3;1}$ | ||||
| 4-12 | 46 | 0 | blowup of 2-33 in the disjoint union of two exceptional lines of the blowup | * | 3-30 | yes | yes | 0 | $\mathbb{G}_{\mathrm{a}}^4\rtimes(\mathrm{GL}_2\times\mathbb{G}_{\mathrm{m}})$ | ||||
| 4-13 | 26 | 0 | blowup of 3-27 in a curve of degree $(1,1,3)$ | 3-27, 3-31 | yes | yes | 1 |
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