Tag: Changho Han

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.

To construct a moduli space which is itself a compactification of a given moduli space, one needs to enlarge the class of objects in consideration (e.g. adding certain singular curves to the class of smooth curves). After a brief review of the compactifications of the moduli of elliptic curves, I will generalize into looking at various compactifications of the moduli of K3 surfaces with nonsymplectic cyclic actions, and then discuss how those compactifications are birationally related to each other. As an application, I will apply this framework into Kondo’s moduli space of sextic K3 surfaces with Z/3Z action. Results come from joint works (in progress) with Valery Alexeev, Anand Deopurkar, and Philip Engel.

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, and Satake used the periods from Hodge theory to compactify the same moduli space. After a brief review of the elliptic curve case (how those notions are the same), I will generalize into looking at various compactifications of Kondo’s moduli space of lattice polarized K3 surfaces (which are of degree 6) with nonsymplectic Z/3Z group action; this involves periods and genus 4 curves by Kondo’s birational period map in 2002. Then, I will extend Kondo’s birational map to describe birational relations between different compactifications by using the slc compactifications (also known as KSBA compactifications) of moduli of surface pairs. The main advantage of this approach is that we obtain an explicit classification of degenerate K3 surfaces, which is used to find geometric meaning of points parametrized by Hodge-theoretic compactifications. This comes from joint works (in progress) with Valery Alexeev, Anand Deopurkar, and Philip Engel.