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In this paper, we study the moduli space of quasi-polarized complex K3surfaces of degree 6 and 8 via geometric invariant theory. The general membersin such moduli spaces are complete intersections in projective spaces and wehave natural GIT constructions for the corresponding moduli spaces and we showthat the K3 surfaces with at worst ADE singularities are GIT stable. We give aconcrete description of boundary of the compactification of the degree 6 casevia the Hilbert-Mumford criterion. We compute the Picard group viaNoether-Lefschetz theory and discuss the connection to the Looijenga'scompactifications from arithmetic perspective. One of the main ingredients isthe study of the projective models of K3 surfaces in terms of Noether-Lefschetzdivisors.

Moduli space of quasi-polarized K3 surfaces of degree 6 and 8