Resumen de The enumerative geometry of cubic hypersurfaces: point and line conditions

Mara Belotti, Alessandro Danelon, Claudia Fevola, Andreas Kretschmer

• The set of smooth cubic hypersurfaces in Pn is an open subset of a projective space. A compactification of the latter which allows to count the number of smooth cubic hypersurfaces tangent to a prescribed number of lines and passing through a given number of points is termed a 1–complete variety of cubic hypersurfaces, in analogy with the space of complete quadrics. Imitating the work of Aluffi for plane cubic curves, we construct such a space in arbitrary dimensions by a sequence of five blow-ups. The counting problem is then reduced to the computation of five total Chern classes. In the end, we derive the desired numbers in the case of cubic surfaces.