Constructing faithful tropicalisations of plane quartics using linear modifications

Any abstract non-hyperelliptic curve of genus $3$ is naturally embedded as a plane quartic via its canonical embedding. In other words, the moduli space of plane quartics maps surjectively onto the non-hyperelliptic locus of the moduli space of genus $3$ curves. It was proved by Brodsky-Joswig-Morrison-Sturmfels, that this does not hold true for the naive definition of the moduli space of tropical plane quartics. In particular, they classified which abstract tropical genus $3$ curves can be realised as a tropical quartic in $\mathbb{R}^2$. We fix this defect of tropical moduli spaces by considering quartics in tropical planes, which themselves are embedded in higher dimension. More precisely, we construct plane embeddings for any abstract tropical non-hyperelliptic genus $3$ curve using modifications. This is joint work in progress with Hannah Markwig, Yue Ren and Ilya Tyomkin.