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

TL;DR

This paper extends the concept of enriched counts from classical enumerative geometry to the 28 bitangents of a smooth plane quartic, addressing challenges due to non-orientability of the associated vector bundle.

Contribution

It introduces a novel approach using a fixed line at infinity to obtain enriched counts of bitangents, overcoming the non-orientability obstacle.

Findings

Enriched counts depend on the geometry relative to the fixed line.

New methods are developed for non-orientable vector bundles.

The approach generalizes $ ext{A}^1$-enumerative geometry techniques.

Abstract

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 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.