Reconstructing general plane quartics from their inflection lines

TL;DR

This paper proves that a general plane quartic can be uniquely reconstructed from its inflection lines, establishing a one-to-one correspondence between the quartic and its inflection line configuration.

Contribution

It demonstrates that the configuration of inflection lines uniquely determines a general plane quartic, providing a new method for reconstructing quartics from geometric data.

Findings

Unique reconstruction of general plane quartics from inflection lines

Inflection line configuration fully determines the quartic

No other quartic shares the same inflection lines as the original

Abstract

Let $C$ be a general plane quartic and let ${\rm Fl}(C)$ denote the configuration of inflection lines of $C$. We show that if $D$ is any plane quartic with the same configuration of inflection lines ${\rm Fl}(C)$, then the quartics $C$ and $D$ coincide.