  In this paper, we study the moduli space of quasi-polarized complex K3
surfaces of degree 6 and 8 via geometric invariant theory. The general members
in such moduli spaces are complete intersections in projective spaces and we
have natural GIT constructions for the corresponding moduli spaces and we show
that the K3 surfaces with at worst ADE singularities are GIT stable. We give a
concrete description of boundary of the compactification of the degree 6 case
via the Hilbert-Mumford criterion. We compute the Picard group via
Noether-Lefschetz theory and discuss the connection to the Looijenga's
compactifications from arithmetic perspective. One of the main ingredients is
the study of the projective models of K3 surfaces in terms of Noether-Lefschetz
divisors.
