Fano's Last Fano

TL;DR

This paper revisits Fano's last Fano 3-fold, providing modern proofs of his construction, and connects it with current classification tools like the Hilbert scheme and Mori-Mukai classification.

Contribution

It offers a modern reinterpretation of Fano's 1949 construction using contemporary algebraic geometry techniques.

Findings

Provides detailed proofs of Fano's construction in modern language

Connects Fano's 3-fold to the Mori-Mukai classification

Uses Hilbert schemes to construct examples

Abstract

In 1949 Fano published his last paper on $3$-folds with canonical sectional curves. There he constructed and described a $3$-fold of the type $X^{22}_3$ in ${\mathbb P}^{13}$ with canonical curve section, which we like to call Fano's last Fano. We report on Fano's construction, providing various (in our opinion missing) proofs, in modern language and trying to use results and techniques available at that time. Then we construct Fano's with modern tools, in particular via the Hilbert scheme of zero cycles on a rational surface; as a consequence we easily point out the corresponding example in the Mori-Mukai classification.