A computational approach to L\"uroth quartics

TL;DR

This paper reviews White and Miller's covariant quartic 4-fold construction and demonstrates its use as a computational tool to identify L"uroth quartics, revealing connections between bitangents and singularities.

Contribution

It introduces a computational approach based on White-Miller's covariant to detect L"uroth quartics and explores their geometric properties.

Findings

The White-Miller quartic 4-fold can identify L"uroth quartics.

28 bitangents correspond to singular points of the White-Miller quartic.

The method provides a practical tool for classifying plane quartics.

Abstract

A plane quartic curve is called L\"uroth if it contains the ten vertices of a complete pentalateral. White and Miller constructed in 1909 a covariant quartic 4-fold, associated to any plane quartic. We review their construction and we show how it gives a computational tool to detect if a plane quartic is L\"uroth. As a byproduct, the 28 bitangents of a general plane quartic correspond to 28 singular points of the associated White-Miller quartic 4-fold.