Résume | In 1981, Tetsuji Shioda proved that, for each integer *m* > 0 prime to 6, the 3*m*^2 lines contained in the *Fermat surface* \Phi_m: z_0^m+z_1^m+z_2^m+z_3^m=0 generate the Picard group of the surface **over Q**, and he conjectured that the same lines also generate the Picard group **over Z**. If true, this conjecture would give us a complete understanding of the Néron--Severi lattice of \Phi_m, leading to the computation of a number of more subtle arithmetical invariants. It was not until 2010 that the first numeric evidence substantiating the conjecture was obtained by Schütt, Shioda, and van Luijk and, in similar but slightly different settings, by Shimada and Takahashi. I will discuss a very simple **purely topological** proof of Shioda's conjecture and try to extend it to the more general so-called *Delsarte surfaces*, where the statement is **not** always true, raising a new open question.
If time permits, I will also discuss a few advances towards the generalization of the conjecture to the (2*d*+1)!! *m*^{*d*+1} projective *d*-spaces contained in the Fermat variety of degree *m* and dimension 2*d*; this part is joint with Ichiro Shimada. |