This lecture note examines the 6 types of “quadrilateral” singularities $J\sb{3,0}$, $Z\sb{1,0}$, $Q\sb{2,0}$, $W\sb{1,0}$, $S\sb{1,0}$, $U\sb{1,0}$: if $X$ is a class of them, $PC(X)$ denotes the set of Dynkin graphs $G$ with components of type $A$, $D$, $E$ having the following property: There is a fibre $Y$ in the versal deformation of a singularity in $X$, such that $Y$ has only rational double points, and $G$ is given as a (disjoint) union of their Dynkin graphs. Quadrilateral singularities are of modality 2, and $PC(X)$ is studied for the case, if $X$ is one of the relevant normal forms [cf. {\it V. I. Arnold}, Invent. Math. 35, 87- 109 (1976; Zbl 0336.57022)]. Due to Looijenga, $PC(X)$ can be studied using the lattice embedding of the associated root-lattice into the even unimodular lattice with signature (19,3). Using Nikulin’s results for such embeddings, this gives a possibility to determine whether or not $G$ belongs to $PC(X)$. The book under review gives a systematic treatment of all cases; technical tools are the root systems $A,\dots,F$ as well as the nonreduced root systems $BC$ arising in the relevant constructions. Using Dynkin graphs (in a terminology slightly different from the standard one), the description of $G \in PC(X)$ (in the cases of $X = J\sb{3,0}$, $Z\sb{1,0}$, $Q\sb{2,0})$ is given by the following theorem: $G$ belongs to $PC(X)$ iff it is in a list of exceptions or can be obtained by applying elementary or “tie transformations” (in the sense, studied by the author in a previous paper) twice to some of a certain list of essential basic Dynkin graphs and if $G$ contains no short root. This theorem can be interpreted in the language of elliptic $K3$-surfaces: It describes the possible combinations of singular fibres on $K3$-surfaces with a singular fibre of type $I\sp*\sb 0$ (in Kodaira’s notation). ‒ The remaining cases of $X = W\sb{1,0}$, $S\sb{1,0}$, $U\sb{1,0}$ require further efforts. Appearing graphs can be characterized by additional rules (presence of “obstruction components” and/or “dual elementary transformations”). The book starts with an introduction to quadrilateral singularities: In chapter 1 Looijenga’s results (which are basic to reduce the problem to lattice embeddings) are reviewed. After an introduction to lattices, another section is devoted to a theory of root systems, adapted to the situation considered later. Further, the technical tools for manipulating graphs are introduced, followed by a section, where conditions are given for a Dynkin graph $G$ to be in $PC(X)$. The chapter concludes explaining Coxeter-Vinberg graphs associated with hyperbolic spaces. Chapter 2 deals with the first three types of quadrilateral singularities, as indicated above, whereas chapter 3 and 4 are devoted to the study of the cases $X = W\sb{1,0}$, $S\sb{1,0}$, $U\sb{1,0}$ respectively. In an appendix, similar questions for plane sextic curves are considered: This is a revised version of the authors’ earlier paper [in Singularities, Proc. IMA Particip. Inst. Conf., Iowa City 1986, Contemp. Math. 90, 295-316 (1989; Zbl 0698.14023)], where the above methods are applied to study conditions for a Dynkin graph to correspond to a configuration of $A,D,E$-singularities on a sextic curve. This book gives insight to deep properties of deformations of a class of bimodal singularities. The author points out essential ideas stemming from a discussion at Oberseminar Brieskorn (University of Bonn), especially from {\it F. J. Bilitewski}, who considered several of the cases treated here, already. The methods employed may be useful to study other types of singularities as well.
Reviewer: M.Roczen (Berlin)