Smooth models of singular \(K3\)-surfaces (Q1733573)

Smooth quartic surfaces in \(\mathbb{P}^3\) are \(K3\)-surfaces, e.g, the Fermat quartic \(\Phi_4: z_0^4+z_1^4+z_2^4+z_3^4=0\in\mathbb{P}^3\). Define a \textit{smooth spatial model} of a \(K3\) surface \(X\) to be an embedding \(X\hookrightarrow \mathbb{P}^3\) defined by a very ample line bundle of degree \(4\). Two models are said to be \textit{projectively equivalent} if so are thier images. The main result of this article is the following Theorem 1. Up to projective equivalence, there are three smooth spatial models \(\Phi_4\hookrightarrow \mathbb{P}^3\)s of the Fermat quartic. They are \(X_{48},\,X_{56},\) and \(\tilde{X}_{56}\). Proofs are given for singular \(K3\) surfaces, i.e., those of the maximal Picard number \(20\). Such a \(K3\) surface is determined by its transcendental lattice \(T\), which is a positive definite even lattice of rank \(2\). Denote by \(X=X(T)\) and \(T=[a,b,c]\) for the lattice \(T=\mathbb{Z}u+\mathbb{Z}v,\, u^2=a, v^2=c,\, u\cdot v=b\). Theorem 1 is formally incorprated in the following Theorem 2. Let \(T\) be a positive definite even lattice of rank \(2\), and assume that \(48\leq \det(T)\leq 80\) and \(T\neq [4,0,16]\). Then, up to projective equivalence, any smooth spatial model \(X(T)\hookrightarrow \mathbb{P}^3\) is one of those listed in Table 1 of the article. There are several corollaries, a prototypical one is: \(\bullet\) \(48\) is the minimal discriminant of a singular \(K3\)-surface admitting a smooth spatial model, and Schur's quartic \(X_{64}\) is the only one minimizing this discriminant. Proofs are mostly rest on a detailed study of the lattice \(S:=h^{\perp}\subset NS(X)\) where \(h\) is the polarization. This approach was earlier used by \textit{S. Kondo} [Duke Math. J. 92, No. 3, 593--603 (1998; Zbl 0958.14025)], and others. As a by-product of this approach, a correlation between the discriminant of a singular \(K3\) surface and the number of lines in its models is observed. Based on this observation, a \(K3\) quartic surface with \(52\) lines and singular points, and other examples with many lines or models are constructed.