The enumerative geometry of cubic hypersurfaces: point and line conditions

TL;DR

This paper develops a method to count smooth cubic hypersurfaces with specific point and line conditions by constructing a compactified moduli space and computing associated Chern classes, advancing enumerative geometry techniques.

Contribution

It introduces a new compactification of the moduli space of cubic hypersurfaces in arbitrary dimensions using blow-ups, enabling precise counting of geometric configurations.

Findings

Constructed a 1-complete variety of cubic hypersurfaces via five blow-ups.

Reduced the counting problem to computing five Chern classes.

Successfully computed the number of cubic surfaces satisfying given conditions.

Abstract

In order to count the number of smooth cubic hypersurfaces tangent to a prescribed number of lines and passing through a given number of points, we construct a compactification of their moduli space. We term the latter a $1$--\textit{complete variety of cubic hypersurfaces} in analogy to the space of complete quadrics. Paolo Aluffi explored the case of plane cubic curves. Starting from his work, we construct such a space in arbitrary dimension by a sequence of five blow-ups. The counting problem is then reduced to the computation of five Chern classes, climbing the sequence of blow-ups. Computing the last of these is difficult due to the fact that the vector bundle is not given explicitly. Identifying a restriction of this vector bundle, we arrive at the desired numbers in the case of cubic surfaces.