Changho Han, Trigonal Curves and Associated K3 Surfaces

K3 surfaces, as a generalization of elliptic curves, have a rich amount of geometric properties. Recalling that elliptic curves are double covers of rational curves branched over 4 distinct points, there are K3 surfaces that are cyclic triple covers of rational surfaces; Artebani and Sarti classified such generic K3 surfaces depending on lattice invariants. Such K3 surfaces admit Kulikov and KSBA degenerations, each leading to toroidal and KSBA compactifications of the moduli spaces of such K3 surfaces. As joint works in progress with Valery Alexeev, Anand Deopurkar, and Philip Engel, I will explain how to use trigonal curves (triple covers of rational curves) to obtain aforementioned degenerations, leading to more explicit understandings of boundaries of those compactifications: such as classifications of generic members and the dimensions.