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There exists a projective $K3$ surface that is not a quartic.
Proof. The moduli space of quartics is irreducible, while the moduli space of projective $K3$s has countably many irreducible components.
There exists a $K3$ surface that is not a quartic.
Proof. The moduli space of quartics is irreducible, while the moduli space of $K3$s has countably many irreducible components.
