Compact moduli space of plane curves via K-stability

We study the K-stability compactification of moduli of log Fano pairs (P^2, aC) where C is a smooth plane curve of degree at least 4. We show that when a is small, the K-moduli compactification is isomorphic to the GIT moduli space of plane curves. We establish a framework of wall-crossing behaviors of these K-moduli spaces as a increases. Specifically, we show that the first wall-crossing of these K-moduli spaces are weighted blow-ups. We describe all wall-crossings for degree 4,5 and 6. We also relate the final K-moduli spaces to Hacking's moduli spaces. This is joint work in progress with K. Ascher, K. DeVleming, and P. Gallardo.