论文标题
立方曲面的枚举几何形状：点和线条条件
The enumerative geometry of cubic hypersurfaces: point and line conditions
论文作者
论文摘要
为了计算与规定的线路数量相切的平滑立方超曲面的数量并通过给定数量的点，我们构造了其模量空间的紧凑型。我们将后者称为$ 1 $ - \ textIt {完整的立方横向表面}，类似于完整四边形的空间。 Paolo Aluffi探索了平面立方曲线的情况。从他的工作开始，我们通过五个爆炸的顺序构建了一个任意维度的空间。然后，计数问题将减少为五个Chern类的计算，从而攀登爆炸的顺序。由于未明确给出矢量束的事实，因此很难计算其中的最后一个。确定对矢量束的限制，我们在立方表面的情况下达到了所需的数字。
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.
