[For the entire collection see Zbl 0668.00006.]
The object of this article is to determine the possible combinations G of rational singularities on a reduced sextic plane curve. By considering the double cover of the plane ramified over the sextic, and the desingularization of this surface, we obtain a K3 surface X. So the question is now a logic continuation of a series of papers by the author and is very similar to the same problem for quartic surfaces [Invent. Math. 87, 549-572 (1987; Zbl 0612.14035)].
The results proved are complete if the sum r of the Milnor numbers of the singularities is \(\leq 14\). For \(r=15, 16\) or 17, the root lattice Q of G must satisfy certain arithmetical restrictions. - The technique of the proof is now almost classical. The root lattice Q embeds in the Picard group of X. A precise converse result is established as an easy consequence of the properties of the period map for K3 surfaces. The problem is reduced in a new one in lattice theory. The restrictions on Q (local: in p-adic fields) are used to construct the expected global embedding of Q in the Picard group.
The possible Dynkin diagrams are obtained by “elementary transforms” (the combinatorial process which gives the subroot systems), repeated twice, from four “maximal” lattices. A nice and useful remark is that these four lattices are closely related to the four kinds of positive even lattices of rank 17 and discriminant 2. - Another result is announced, without proof, and a corollary is a complete classification of sextics singularities with Milnor number 15 (without the restrictions mentioned above).