A classical result in the theory of algebraic surfaces is that the blow-up of a finite number of points in the projective plane is a del Pezzo surface if and only if the number of points is bounded by 8 and if no three are collinear, no six on the same conic and no eight on the same cubic, singular at one of the eight points. In particular, the case of six points gives all possible smooth cubic surfaces, studied by Segre.

In this talk, I would like to describe the situation in higher dimension. The case of points being an easy exercise, I will focus on the case of blow-up of smooth curves in the projective space, and describe when the variety obtained is Fano, or weak-Fano. This is a joint work with Stéphane Lamy.