The influence of height on degenerations of algebraic surfaces of type $K3$

The authors announce the conjecture that a family of $K3$ surfaces the Artin height of whose generic fiber is greater than $2$ does not degenerate; they prove this conjecture for surfaces of degree $2$. As a corollary it is shown that a family of supersingular $K3$ surfaces does not degenerate; i.e., its variety of moduli is complete.
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