Normal algebraic surfaces with trivial tricanonical divisors
A normal projective algebraic surface \(S\) with at worst quotient singularities is called a log Enriques surface (LES) if \(H^1 ({\mathcal O}_S) =0\) and some multiple of \(K_S\) is trivial. The smallest such multiple \(I\) is defined as the index of \(S\). Consider the canonical Galois cover \(\pi:Y=\text{proj} \bigoplus^{I-1}_{i=0} {\mathcal O}_S (iK_S)\to S\) with Galois group \(\langle \sigma \rangle\) of order \(I\). \(Y\) is either an abelian or K3 surface with at worst Du Val singularities. \(S\) is said of actual type \(A_n\) \((D_n)\) if \(Y\) has exactly one Du Val points of the corresponding type. The article under review studies rational LES whose canonical cover has a singular point of type \(A_{17}\) or \(D_{17}\) and of index 3. Using the unique algebraic K3 surface of Picard number 20 and discriminant 3 constructed by Shioda and Inose [see ``On the most algebraic K3 surfaces and the most extremal log Enriques surfaces'' by \textit{K. Oguiso} and \textit{D.-Q. Zhang}, in Am. J. Math. 118, No. 6, 1277-1297 (1996; Zbl 0889.14016)], the author constructs three index 3 rational LES of actual type \(A_{17}\) by contracting the three pairs of rational curves located at the ends of the \(D_{19}\) graph in the K3 surface followed by taking quotient under an order 3 automorphism. It is proved that any rational LES of index 3 of type \(A_{17}\) (i.e.; whose canonical cover has an \(A_{17}\) singularity) must be isomorphic to one of the three examples by showing that there are two smooth rational curves on the minimal resolution \(X\) of the canonical cover \(Y\) jointly with the graph corresponding to the \(A_{17}\) singularity form a \(D_{19}\) graph and \(X\) must be the unique Shioda-Inose surface. By results from early works by the author [``Normal algebraic surfaces with trivial two or four times of the canonical divisors''] and the author and \textit{K. Oguiso} [``K3 surfaces with order 5 automorphisms''] the index of a rational LES of type \(A_{17}\) has to be one of \(2, 3, 4, 5, 6, 12\). The index \(2, 4, 5\) cases were studied. The canonical cover \(Y\) of an index 6 LES of type \(A_{17}\) is also that (by taking quotient of \(Y\) by \(\langle \sigma^2 \rangle\) where \(\sigma\) is of order 6) of an index 3 LES \(S\) of the same type which is now completely understood. Comparing the fixed points of a K3 surface under an order 6 automorphism and that on the minimal resolution of three examples constructed before under an order 3 automorphism, the author concludes that \(I\) can not be 6 (hence not 12 either). It is also proved that a type \(D_{17}\) rational LES does not exist.
scientific article; zbMATH DE number 1117532
scientific article