Tie transformations of Dynkin graphs and singularities on quartic surfaces
This article is the continuation of my previous one in Invent Math. 87, 549-572 (1987; Zbl 0612.14035). We continue to study possible combinations of rational double points on quartic surfaces in the projective space \(\mathbb{P}^3\) of dimension 3. Now in the cited paper I proposed a certain operation on Dynkin graphs, which I called an elementary transformation. This notion was natural and simple. We would like to propose another procedure, called a tie transformation. Performing such a transformation can increase the number of vertices in the Dynkin graph by one. In this, it differs from elementary transformations, since an elementary transformation never makes the number of vertices larger. Because of this property of tie transformations we can treat many interesting examples of K3 surfaces whose Picard number is maximal 20. A systematic description for possible combinations of rational double points on quartic surfaces by these two kinds of transformations of Dynkin graphs is given. Indeed, the following is a part of the main results: Theorem 0.2. Any Dynkin graph without vertices corresponding to short roots obtained from one of the following 9 basic Dynkin graphs by elementary or tie transformations repeated twice (Four kinds of combinations - i.e., ``elementary'' twice, ``tie'' twice, ``elementary'' after ``tie'', and ``tie'' after ``elementary'' - are all permitted.) describes the combination of singularities of a quartic surface in the projective space of dimension 3 with only rational double points as singularities. - The basic Dynkin graphs: \(A_{11} + E_6\), \(2D_8 + B_1\), \(D_{16} + B_ 1\), \(2E_ 8 + B_1\), \(A_{15} + B_2\), \(2E_7 + B_3\), \(D_{12} + B_5\), \(E_8 + B_9\), \(B_{17}\). Theorem 0.4'. Let \(G=\sum a_ kA_ k +\sum b_ lD_ l +\sum c_ mE_ m \) (a finite sum) be a Dynkin graph with components of type A, D or E only. Set \(r=\sum a_ kk +\sum b_{\ell}\ell +\sum c_ mm \). Then the following conditions (A) and (B) are equivalent: (A) There exists a normal quartic surface in the projective space of dimension 3 whose combination of singularities just agrees with G and moreover one of the following conditions \(<1>, <2>, <3>\) holds for the root lattice \(Q=Q(G)\) of type G (by d(Q) we denote the discriminant of Q): \(<1>\quad r=18,\) and for every prime number \(p,\) \(\epsilon_ p(Q)=1\). - \(<2>\quad r=17,\) and for every prime number \(p,\) \(d(Q)\not\in {\mathbb{Q}}_ p^{*2}\) or \(\epsilon_ p(Q)=1\). - \(<3>\quad r\leq 16.\) (B) One of the following (B-1), (B-2) holds: (B-1) G is a Dynkin graph obtained from one of the 9 basic Dynkin graphs in theorem 0.2 \(<1>\) by elementary transformations repeated twice; - \(<2>\) by an elementary transformation following after a tie transformation; - or \(<3>\) by a tie transformation following after an elementary transformation such that it has no vertex corresponding to a short root. (B-2) G is a Dynkin graph obtained from one of the following 11 sub-basic Dynkin graphs by one elementary transformation such that it has no vertex corresponding to a short root. - The sub-basic Dynkin graphs: \[ \begin{alignedat}{4} &A_{11} + A_5 + B_2,\quad & &A_9 + A_6 + A_1 + B_2,\quad & &2A_8 + B_2,\quad & &2A_7 + 2A_1 + B_2, \\ &A_{10} + A_6 + a_2,\quad & &2a_9,\quad & &2A_7 + A_3 + A_1,\quad & &A_7 + A_6 + A_3 + A_2, \\ &A_7 + 3A_3 + A_2,\quad & &3A_6,\quad & &6A_3, \end{alignedat} \] In the above \(\epsilon_ p(Q)=\pm 1\) denotes the Hasse invariant of the inner product space \(Q\otimes {\mathbb{Q}}_ p\) over the p-adic number field \({\mathbb{Q}}_ p\). Note that in the case where the number of vertices in the Dynkin graph G is less that or equal to 16, possible combinations of rational double points on quartic surfaces are completely described by the above theorem.
rational double points on quartic surfaces
K3 surfaces whose Picard number is maximal 20
scientific article; zbMATH DE number 4091649
scientific article