Fano's Last Fano

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.