Classification of unstable quartic plane curves

Consider the action by change of coordinates in the space of quartic plane curves. We classify the unstable points of this action through a stratification by irreducible, smooth and locally closed subvarieties. Concretely, we show that these varieties parametrize quartic plane curves with specific singularities, we compute their dimensions and closure. In addition, we show that the space of unstable points consists of two irreducible components which are the closure of two of the strata.