Compactifications of the moduli space of plane quartics and two lines
The aim of the paper under review is to study the moduli space of \(K3\) surfaces of degree \(2\) that are double covering of the projective space branched at the union of a quartic curve and two lines. The construction here is given in two ways that are finally turned to coincide. The first part is devoted to study the space of semi-/stable conditions for a tuple of a hypersurface and \(k\) hyperplanes in the projective space as a generalisation of the result obtained by the first two authors. Applying this, they describe the moduli space as the GIT quotient for triples of a quartic curve and two lines. As a result, they obtain an isomorphic period map constructed in Section 3.6. In the second part, they construct the moduli space as that of polarised \(K3\) surfaces by the lattice \(M\) presented in Notation 3.3. The strategy is to first describe the period map for generic intersections, and then, by using the method of the normalised-lattice polarisation for degree-5 pairs due to \textit{R. Laza} [Trans. Am. Math. Soc. 361, No. 4, 2109--2161 (2009; Zbl 1174.14004)] they construct a period mapping to be a morphism, specifically to describe the extension at non-generic intersections.
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