An enriched count of the bitangents to a smooth plane quartic curve

Recent work of Kass–Wickelgren gives an enriched count 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 answer is computed as degree Euler class relatively orientable vector bundle. Here, we consider closely related problem 28 bitangents to plane quartic. However, it turns out relevant bundle not and new ideas are needed produce counts. We introduce fixed “line at infinity,” which leads counts depend their relative quartic this distinguished line.