Algebraic Surfaces with Log-Terminal Singularities and nef Anticanonical Class and Reflection Groups in Lobachevsky Spaces. I (Basics of the Diagram Method)

Abstract

In our preprint: "Algebraic surfaces with log-terminal singularities and nef anticanonical class and reflection groups in Lobachevsky spaces", Preprint Max-Planck-Institut für Mathematik, Bonn, (1989) MPI/89-28 (Russian), we had extended the diagram method to algebraic surfaces with nef anticanonical class (before, it was applied to Del Pezzo surfaces). The main problem here is that Mori polyhedron may not be finite polyhedral in this case. In the §4 of this preprint we had shown that this method does work for algebraic surfaces with nef anticanonical class and log terminal singularities. Recently, using in particular these our results, Valery Alexeev (Boundedness and K^2 for log surfaces, Preprint (1994), alg-geom/942007), obtained strong and important results for surfaces of general type with semi-log canonical singularities. The reason is that Del Pezzo surfaces (the diagram method is especially effective for them) and surfaces of general type "are connected" through surfaces with numerically zero canonical class. This was the main reason that we decided to make this English translation of the first part (§1--§3) of our preprint above with some extensions (where we give omitted proofs) and minor corrections. This part contains basics of the diagram method for generalized reflection groups in Lobachevsky spaces and for surfaces with nef anticanonical class.