We enrich the classical count that there are two complex lines meeting four lines in space to an equality of isomorphism classes of bilinear forms. For any field $k$, this enrichment counts the number of lines meeting four lines defined over $k$ in $\mathbb{P}^3_k$, with such lines weighted by their fields of definition together with information about the cross-ratio of the intersection points and spanning planes. We generalize this example to an infinite family of such enrichments, obtained using an Euler number in $\mathbb{A}^1$-homotopy theory. The classical counts are recovered by taking the rank of the bilinear forms. In the appendix, the condition that the four lines each be defined over $k$ is relaxed to the condition that the set of four lines being defined over $k$.

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$P^3$ 中与四行相交的行的算术计数

我们丰富了经典计数，即有两条复线在空间中与四条线相遇，从而使双线性形式的同构类相等。对于任何字段 $k$，此扩充计算与 $\mathbb{P}^3_k$ 中 $k$ 上定义的四行相交的行数，这些行按其定义字段以及交叉比信息加权的交点和跨越平面。我们将这个例子推广到一个无限的此类丰富系列，使用 $\mathbb{A}^1$-同伦理论中的欧拉数获得。通过取双线性形式的秩来恢复经典计数。在附录中，四行每行定义在 $k$ 上的条件放宽到四行集定义在 $k$ 上的条件。