Plane quartics with at least 8 hyperinflection points

TL;DR

This paper demonstrates that smooth plane quartic curves with at least 8 hyperinflection points can be uniquely reconstructed from their inflection lines, extending known results and addressing cases in positive characteristic.

Contribution

It proves that such quartics are determined by their inflection lines, even in positive characteristic with specific exceptions, advancing understanding of curve reconstruction.

Findings

Reconstruction of quartics with ≥8 hyperinflection points from inflection lines

Extension of results to positive characteristic fields with exceptions

Identification of characteristic 13 as an exception in positive characteristic

Abstract

A recent result shows that a general smooth plane quartic can be recovered from its 24 inflection lines and a single inflection point. Nevertheless, the question whether or not a smooth plane curve of degree at least 4 is determined by its inflection lines is still open. Over a field of characteristic 0, we show that it is possible to reconstruct any smooth plane quartic with at least 8 hyperinflection points by its inflection lines. Our methods apply also in positive characteristic, where we show a similar result, with two exceptions in characteristic 13.