Yuna Herlédan

It is a classical fact that vector spaces of finite dimension split into direct sums of as many vector lines as their dimensions by choosing a basis. In the case of vector bundle, that is families of vector spaces over a geometric space, it depends on the geometry of the basis. Even over nice spaces such as the projective plane a vector bundle may not even contain a line bundle. Over the projective line, at least, it is always true, so we could think of looing at the restrictions of the bundle to lines. Luckily, projectives spaces are filled with lines. I will expose some of the benefits of this method as well as Horrock's splitting criterion: a vector bundle with the cohomology of a direct sum of line bundles splits.