\(\mathbb{A}^1\) curves on log \(K3\) surfaces
The aim of the article under review is to study (the existence of) \(\mathbb{A}^1\) curves on genuine log \(K3\) surfaces of type II. A log \(K3\) surface of type II is by definition a pair \((X,\, D)\) of smooth projective rational surface \(X\) and a divisor \(D\) on \(X\) with logarithmic geometric genus \(1\), and logarithmic Kodaira dimension and logarithmic irregularity are zero. In the paper, it is assumed that \((X,\, D)\) is genuine: namely, the divisor \(D\) is numerically equivalent to the anticanonical divisor of \(X\), and the global sections of the sheaf of logarithmic \(1\)-forms on \(X\) are all trivial. In the first step, a classification of genuine log \(K3\) surfaces of type II is obtained by giving an explicit description of the divisor \(D\) such that a pair \((X,\, D)\) is a genuine log \(K3\) surface of type II. There are five classes C0 to C4 up to log isomorphism. With this classification, the main theorem is given that a genuine log \(K3\) surface of type II \((X,\, D)\) contains countably-many \(\mathbb{A}^1\) curves if and only if \((X,\, D)\) is log isomorphic to C0, C1, C2, or C3. Lastly, as an application, some examples of genuine log \(K3\) surfaces of type II that contain only finitely many \(\mathbb{A}^1\) curves are given. The reviewer wonders without the assumption ``genuine'', if it is possible to classify all of log \(K3\) surfaces of type II, and to study the existence of \(\mathbb{A}^1\) curves on them.
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