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 degree $22$ in a projective space of dimension $13$ with canonical curve section, which we like to call Fano's last Fano. We report on Fano's construction and we provide various (in our opinion missing) proofs in modern language. Moreover, we try to use results and techniques available at that time. After that we construct Fano's last Fano 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.