The enumerative geometry of cubic hypersurfaces: point and line conditions

We are interested in counting cubic hypersurfaces in projective n-space tangent to enough many points and lines. Paolo Aluffi explored the case for plane cubic curves. Starting from his work we construct a 1-complete variety of cubic hypersurfaces by a sequence of five blow-ups over the space parametrizing the cubics. The problem is then reduced to compute five Segre classes by climbing the sequence of blow-ups. This is an ongoing project with Mara Belotti, Alessandro Danelon e Andreas Kretschmer.