Recent work of Kass–Wickelgren gives an enriched count of the 27 lines on a smooth cubic surface over arbitrary fields. Their approach using \(\mathbb {A}^1\)-enumerative geometry suggests that other classical enumerative problems should have similar enrichments, when the answer is computed as the degree of the Euler class of a relatively orientable vector bundle. Here, we consider the closely related problem of the 28 bitangents to a smooth plane quartic. However, it turns out that the relevant vector bundle is not relatively orientable and new ideas are needed to produce enriched counts. We introduce a fixed “line at infinity,” which leads to enriched counts of bitangents that depend on their geometry relative to the quartic and this distinguished line.

Kass–Wickelgren的最新工作丰富了任意场在光滑立方体表面上的27条线的数量。他们使用\（\ mathbb {A} ^ 1 \）-枚举几何的方法表明，当将答案计算为相对可定向的矢量束的Euler类的程度时，其他经典的枚举问题也应具有相似的充实度。在这里，我们考虑将28位切线问题与光滑四次方紧密相关的问题。然而，事实证明相关的向量束不是相对定向的，需要新的思想来产生丰富的计数。我们引入了一条固定的“无穷远线”，这导致了相对于四次方线和这条独特线而言，取决于其几何形状的双切线的丰富计数。