Modèles canoniques plongés. I. (Embedded canonical models. I) (Q1192240)

In higher dimensional geometry resolutions are not unique. In dimension two there are still minimal resolutions, but embedded resolutions are already not unique. Threefold theory suggests to look at embedded canonical models, for which the ambient space has also at most canonical singularities. In this paper toric methods are used to construct such models for singularities, defined by functions which are nondegenerate for their Newton diagrams. The paper first describes the Newton blow-up, the modification defined by the Newton diagram, in terms of an equivariant normalised blow-up, in arbitrary dimensions. The main results concern the surface case; let \(f\to k\) be nondegenerate for its Newton diagram, put \(S=\{f=0\}\), let \(\hat V\to V\) be the Newton blow-up, and let \(\hat S\) be the strict transform of \(S\). The authors prove that \(\hat S\) has only \(A_ k\)- singularities in smooth points of Sing\(\hat V\), and they describe in detail how the surface \(\hat S\) intersects the exceptional set (transversally, which has to be explained in the presence of singular points). This embedded canonical model \(\hat S\subset\hat V\) of \(S\subset V\) is minimal for toric morphisms. As example the authors work out the case of \(E_ 6\), which has five terminal embedded resolutions, related by flops, but the toric canonical model is unique.