ncts

Abstract: This talk is based on the preprint arxiv:2502.04301. Moduli spaces of K3 surfaces have long been studied, beginning with the proof of the Torelli theorem for K3s in the 70s. Moreover, Pjatecki—Shapiro and Shafarevich showed that the moduli space of K3 surfaces with an ample divisor of fixed positive self-intersection was quasi-projective. The most well-known compactificaton of this moduli space is still probably the Baily-Borel compactification, whose boundary components consist of curves and points corresponding to Type II and Type III degenerations respectively. Meanwhile, the GIT compactification is able to explicitly describe elements belonging to its boundary components. In this talk we consider the simplest Type II degenerations, called Tyurin degenerations. In particular, we explicitly describe degenerations which correspond to each of the Type II boundary components of the Baily-Borel compactification of the moduli space of K3 surfaces of degree 4, we also describe their stable models and relate this to Mumford's toroidal compactifications. To conclude, we will discuss ongoing work to prove analogous theorems in higher degree via computer algorithms.