×

Modèles canoniques plongés. I. (Embedded canonical models. I). (French) Zbl 0772.14008

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.

MSC:

14E15 Global theory and resolution of singularities (algebro-geometric aspects)
14J30 \(3\)-folds
14J17 Singularities of surfaces or higher-dimensional varieties
32S45 Modifications; resolution of singularities (complex-analytic aspects)
14B05 Singularities in algebraic geometry
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] BOURBAKI, N., Algebre commutative, chap. 7.
[2] DANILOV V. I., The geometry of toric varieties, Russian Math. Surveys, 33 (1978), 97-154; Uspeki Mat. Nauk, 33 (1978), 85-134 · Zbl 0425.14013
[3] Gonzalez-Sprinberg, G., Quelques descriptions de desingularisations plongees d surface, Prepublication de lnstitut Fourier no. 123. · Zbl 0766.14007
[4] LEJEUNE-JALABERT, M., Desingularisation explicite des surfaces quasi-homogene dans C3, Nova acta Leopoldina NF 52 Nr., 240 (1980), 139-160.
[5] MORI, S., Flip theorem and the existence of minimal models for 3-folds, Journa of the A. M. S., 1, no. l (1988), 117-253. · Zbl 0649.14023
[6] ODA, T., Lectures on Torus Embeddings and Applications, Tata Inst. of Fund Research, 58, Springer, 1978. · Zbl 0417.14043
[7] ODA, T., Convex Bodies and Algebraic Geometry, An Introductionto the Theor of Toric Varieties, Ergebnisse der Math., 15, Springer Verlag, 1988. · Zbl 0628.52002
[8] OKA, M., On the resolution of the hypersurface singularities in Complex Analyti Singularities, Adv. Studies in Pure Math., 8 (1986), 405-436. · Zbl 0622.14012
[9] REID, M., Canonical 3-folds, Journees de Geometric Algebrique d’Angers (1979), Noordhoff (1980), 273-310 · Zbl 0451.14014
[10] KEMPF, G., KNUDSEN, F., MUMFORD, D. AND SAINT DONAT, B., Toroida Embeddings I, Lecture Notes in Mathematics, 339, Springer, 1973. · Zbl 0271.14017
[11] VARCHENKO, A. N., Zeta function of monodromy and Newton’s diagram, Invent Math., 37 (1976), 253-262. · Zbl 0333.14007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.