K-stability and birational models of moduli of quartic K3 surfaces

Abstract

We show that the K-moduli spaces of log Fano pairs $({\mathbb{P}}^3,cS)$ where $S$ is a quartic surface interpolate between the GIT moduli space of quartic surfaces and the Baily-Borel compactification of moduli of quartic K3 surfaces as $c$ varies in the interval $(0,1)$. We completely describe the wall crossings of these K-moduli spaces. As the main application, we verify Laza-O'Grady's prediction on the Hassett-Keel-Looijenga program for quartic K3 surfaces. We also obtain the K-moduli compactification of quartic double solids, and classify all Gorenstein canonical Fano degenerations of ${\mathbb{P}}^3$.

56 pages, comments are very welcome. v2: to appear in Invent. Math

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