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We prove that the universal family of polarized K3 surfaces of degree 2 can be extended to a flat family of stable KSBA pairs over the toroidal compactification associated to the Coxeter fan. One-parameter degenerations of K3 surfaces in this family are described by integral-affine structures on a sphere with 24 singularities.more » « less
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Abstract The flex divisor$$R_{\textrm flex}$$ of a primitively polarized K3 surface $(X,L)$ is, generically, the set of all points $$x\in X$$ for which there exists a pencil $$V\subset |L|$$ whose base locus is $$\{x\}$$. We show that if $L^2=2d$ then $$R_{\textrm flex}\in |n_dL|$$ with $$ \begin{align*} &n_d= \frac{(2d)!(2d+1)!}{d!^2(d+1)!^2} =(2d+1)C(d)^2,\end{align*}$$where $C(d)$ is the Catalan number. We also show that there is a well-defined notion of flex divisor over the whole moduli space $$F_{2d}$$ of polarized K3 surfaces.more » « less
