Weyl groups and the KKMR decomposition
Following Matsuki, one considers the relative movable cone attached to a Fano threefold and its decomposition into chambers corresponding to minimal models. This is the Kawamata–Kollár–Mori–Reid (KKMR) decomposition. The associated Weyl group is the group of integral lattice symmetries that preserves this chamber decomposition. It therefore records symmetries among minimal models; it is not the automorphism group of the Fano threefold itself.
The labels $\mathrm{A}_n$, $\mathrm{D}_n$, and $\mathrm{E}_n$ give the root-system type of the group. The table lists exactly the 35 families with nontrivial Weyl group. Every family not listed has the trivial group $\{e\}$.
The data comes from Matsuki's original analysis, with corrections and the missing family 4-13 from the addendum.
| ID | $\rho$ | Weyl group | description |
|---|---|---|---|
| 2-2 | 2 | $\mathrm{A}_1$ | double cover of 2-34 with branch locus a $(2,4)$-divisor |
| 2-6 | 2 | $\mathrm{A}_1$ | Verra 3-fold
|
| 2-12 | 2 | $\mathrm{A}_1$ | intersection of 3 $(1,1)$-divisors in $\mathbb{P}^3\times\mathbb{P}^3$
|
| 2-21 | 2 | $\mathrm{A}_1$ | blowup of 1-16 in a twisted quartic |
| 2-32 | 2 | $\mathrm{A}_1$ | divisor on $\mathbb{P}^2\times\mathbb{P}^2$ of bidegree $(1,1)$
|
| 3-1 | 3 | $\mathrm{A}_2$ | double cover of 3-27 with branch locus a divisor of degree $(2,2,2)$ |
| 3-3 | 3 | $\mathrm{A}_1$ | divisor on $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^2$ of degree $(1,1,2)$ |
| 3-7 | 3 | $\mathrm{A}_1$ | blowup of 2-32 in an elliptic curve which is the intersection of two divisors from $|-\frac{1}{2}\mathrm{K}_W|$ |
| 3-9 | 3 | $\mathrm{A}_1$ | blowup of the cone over the Veronese of $\mathbb{P}^2$ in $\mathbb{P}^5$ with center the disjoint union of the vertex and a quartic curve on $\mathbb{P}^2$ |
| 3-10 | 3 | $\mathrm{A}_1$ | blowup of 1-16 in the disjoint union of 2 conics
|
| 3-13 | 3 | $\mathrm{A}_1$ | blowup of 2-32 in a curve $C$ of bidegree $(2,2)$ such that the composition $C\hookrightarrow W\hookrightarrow\mathbb{P}^2\times\mathbb{P}^2\overset{p_i}{\to}\mathbb{P}^2$ is an embedding for $i=1,2$ |
| 3-17 | 3 | $\mathrm{A}_1$ | divisor on $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^2$ of degree $(1,1,1)$ |
| 3-19 | 3 | $\mathrm{A}_1$ | blowup of 1-16 in two non-collinear points |
| 3-20 | 3 | $\mathrm{A}_1$ | blowup of 1-16 in the disjoint union of two lines |
| 3-25 | 3 | $\mathrm{A}_1$ | blowup of 1-17 in the disjoint union of two lines
|
| 3-27 | 3 | $\mathrm{A}_2$ | $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$ |
| 3-31 | 3 | $\mathrm{A}_1$ | blowup of the cone over a smooth quadric in $\mathbb{P}^3$ in the vertex
|
| 4-1 | 4 | $\mathrm{A}_3$ | divisor on $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$ of degree $(1,1,1,1)$ |
| 4-2 | 4 | $\mathrm{A}_1\times\mathrm{A}_1$ | 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 |
| 4-3 | 4 | $\mathrm{A}_1$ | blowup of 3-27 in a curve of degree $(1,1,2)$ |
| 4-4 | 4 | $\mathrm{A}_1$ | blowup of 3-19 in the proper transform of a conic through the points |
| 4-6 | 4 | $\mathrm{A}_2$ | blowup of 1-17 in the disjoint union of 3 lines
|
| 4-7 | 4 | $\mathrm{A}_1$ | blowup of 2-32 in the disjoint union of a curve of degree $(0,1)$ and a curve of degree $(1,0)$ |
| 4-8 | 4 | $\mathrm{A}_1$ | blowup of 3-27 in a curve of degree $(0,1,1)$ |
| 4-10 | 4 | $\mathrm{A}_1$ | $\mathbb{P}^1\times\mathrm{Bl}_2\mathbb{P}^2$ |
| 4-12 | 4 | $\mathrm{A}_1$ | blowup of 2-33 in the disjoint union of two exceptional lines of the blowup |
| 4-13 | 4 | $\mathrm{A}_1$ | blowup of 3-27 in a curve of degree $(1,1,3)$ |
| 5-1 | 5 | $\mathrm{A}_2$ | blowup of 2-29 in the disjoint union of three exceptional lines of the blowup |
| 5-2 | 5 | $\mathrm{A}_1$ | blowup of 3-25 in the disjoint union of two exceptional lines on the same irreducible component |
| 5-3 | 5 | $\mathrm{A}_1\times\mathrm{A}_2$ | $\mathbb{P}^1\times\mathrm{Bl}_3\mathbb{P}^2$ |
| 6-1 | 6 | $\mathrm{A}_4$ | $\mathbb{P}^1\times\mathrm{Bl}_4\mathbb{P}^2$ |
| 7-1 | 7 | $\mathrm{D}_5$ | $\mathbb{P}^1\times\mathrm{Bl}_5\mathbb{P}^2$ |
| 8-1 | 8 | $\mathrm{E}_6$ | $\mathbb{P}^1\times\mathrm{Bl}_6\mathbb{P}^2$ |
| 9-1 | 9 | $\mathrm{E}_7$ | $\mathbb{P}^1\times\mathrm{Bl}_7\mathbb{P}^2$ |
| 10-1 | 10 | $\mathrm{E}_8$ | $\mathbb{P}^1\times\mathrm{Bl}_8\mathbb{P}^2$ |