# Golyshev's periodic table for Fano 3-folds of rank 1

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 | $\mathbb{P}^3$ |

**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 $\mathbb{P}^4$