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arXiv · 1909.05945

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

Hannah Larson·Isabel Vogt
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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.

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Hannah Larson, Isabel Vogt. 2019-09-12. An enriched count of the bitangents to a smooth plane quartic curve. https://arxiv.org/abs/1909.05945

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