Title: Moduli of 'almost K3' stable log surfaces, curves of genus 4, and degree 6 K3 surfaces with nonsymplectic Z/3Z group actions.
Abstract: Observe that any construction of "meaningful" compactification of moduli spaces of objects involve enlarging the class of objects in consideration. For example, Deligne and Mumford introduced the notion of stable curves in order to compactify the moduli of smooth curves of genus g. I will briefly introduce different realizations of smooth curves of genus 4, which gives a birational map between the moduli of alpha-stable curves, moduli of 'almost K3' stable log surfaces (where the underlying surfaces are rational), and the moduli of degree 6 K3 surfaces with nonsymplectic Z/3Z group action. Then, I will describe joint works with Anand Deopurkar that describes the moduli of 'almost K3' stable log surfaces via moduli of curves of genus 4. If time remains, I will introduce observations from the work in progress with Valery Alexeev, Anand Deopurkar, and Philip Engel on relation with the Baily-Borel compactifications of such K3 surfaces.