The set of smooth cubic hypersurfaces in \({{\mathbb {P}}}^n\) 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.

\({{\mathbb {P}}}^n\)中的光滑立方超曲面集是射影空间的开子集。后者的紧化允许计算与规定数量的线相切并通过给定数量的点的光滑立方超曲面的数量，称为 1-完全立方超曲面，类似于完全二次曲面的空间。模仿 Aluffi 对平面三次曲线的工作，我们通过五次放大的顺序构造了这样一个任意维度的空间。然后将计数问题简化为计算五个陈类。最后，我们在立方曲面的情况下推导出所需的数字。