A few weeks ago Sergey Galkin explained a funny fact about the classification of Fano 3-folds of rank 1, due to Golyshev (the fact that is, not the classification, which is due to Mori–Mukai). Namely, if you make a table of these Fano 3-folds as follows
|
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
11 |
| 1 |
$\mathrm{V}_2$ |
$\mathrm{V}_4$ |
$\mathrm{V}_6$ |
$\mathrm{V}_8$ |
$\mathrm{V}_{10}$ |
$\mathrm{V}_{12}$ |
$\mathrm{V}_{14}$ |
$\mathrm{V}_{16}$ |
$\mathrm{V}_{18}$ |
$\mathrm{V}_{22}$ |
| 2 |
$\mathrm{B}_1$ |
$\mathrm{B}_2$ |
$\mathrm{B}_3$ |
$\mathrm{B}_4$ |
$\mathrm{B}_5$ |
| 3 |
|
|
$Q$ |
| 4 |
|
$\mathbb{P}^3$ |
where the rows are indexed by the Fano index $r$, and the columns by the integer $N$, then
\begin{equation}
I=r\psi(N)
\end{equation}
where $\psi(N)=N\prod_{p\,\mid\, N}(1+1/p)$ is the
Dedekind psi function, is a
birational invariant. So whenever the value is
12 in the following table, the Fano variety is rational. You also see the correspondence between $\mathrm{B}_n$ and $\mathrm{V}_{4n+2}$.
|
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
11 |
| 1 |
1 |
3 |
4 |
6 |
6 |
12 |
8 |
12 |
12 |
12 |
| 2 |
2 |
6 |
8 |
12 |
12 |
| 3 |
|
|
12 |
| 4 |
|
12 |
At some point I'd like to do more with the classification of Fano 3-folds, I feel like something similar to superficie.info would work really well for this.
As a quick reminder, here are the descriptions of the varieties in the first table.
- $\mathrm{V}_2$
- sextic hypersurface in $\mathbb{P}(1,1,1,1,3)$
- $\mathrm{V}_4$
- quartic hypersurface in $\mathbb{P}^4$
- $\mathrm{V}_6$
- complete intersection of a quadric and cubic in $\mathbb{P}^5$
- $\mathrm{V}_8$
- complete intersection of 3 quadrics in $\mathbb{P}^6$
- $\mathrm{V}_{10}$
- section of $\mathrm{Gr}(2,5)$ by quadric and codimension 2 hyperplane plane
- $\mathrm{V}_{12}$
- section of $\mathrm{OGr}(5,10)$ by codimension 7 hyperplane
- $\mathrm{V}_{14}$
- section of $\mathrm{Gr}(2,6)$ by codimension 5 hyperplane
- $\mathrm{V}_{16}$
- section of $\mathrm{LGr}(3,6)$ by codimension 3 hyperplane
- $\mathrm{V}_{18}$
- section of $\mathrm{G}_2/P$ by codimension 2 hyperplane
- $\mathrm{V}_{22}$
- hard to describe succinctly
- $\mathrm{B}_1$
- sextic hypersurface in $\mathbb{P}(1,1,1,2,3)$
- $\mathrm{B}_2$
- quartic hypersurface in $\mathbb{P}(1,1,1,1,2)$
- $\mathrm{B}_3$
- cubic hypersurface in $\mathbb{P}^4$
- $\mathrm{B}_4$
- complete intersection of 2 quadrics in $\mathbb{P}^5$
- $\mathrm{B}_5$
- section of $\mathrm{Gr}(2,5)$ by codimension 3 hyperplane
- $Q$
- quadric hypersurface in $\mathbb{P}^4$