2024/10/24 by Belotti, Mara, Danelon, Alessandro, Fevola, Claudia +1
#500 Naturwissenschaften und Mathematik::510 Mathematik::510 Mathematik #algebra #analysis #applications of mathematics #geometry
paper · doi:10.14279/depositonce-21510
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.