THEOREM 6.3. Let g: Tw U be as above. Let W {u E U | Xu has a singularity
pu such that (Xu9 pu) is homeomorphic to (V, p)}. Then g(Tw) W.
g(Tw) C W be Definition 5.1. The rest of the proof of Theorem 6.3 is
the same as the proof of Theorem 5.6 via the changes:
"W" replaces " S " and the condition {Xu has a singularity pu such that
(Xu9 pu) is homeomorphic to (V, p)} replaces the condition {Ku Ku = K0 K0}.
"Tw" replaces "Ta".
Lemma 6.1 replaces Theorem 5.2.
Lemma 6.2 replaces Theorem 4.3 and [8]. The base change a: Tx - T is not
co: DR/ - 0 ' replaces co: 9H g.
6.4. Lef X'.'Y-* T be the germ of a (flat) deformation of the normal
Gorenstein two-dimensional singularity (V, p) with T a reduced analytic space. Then
X has a weak simultaneous resolution if and only if each Vt has a singularity pt such
that (Vv pt) is homeomorphic to (V, p).
This theorem follows immediately from Definition 5.1, Theorem 6.3,
the versality of r: % - U and the fact [52, Theorem 4.6(b), p. 341] that g: Tw - W
is biholomorphic.
1. M. Artin, On isolated rational singularities of surfaces, Amer. J. Math. 88 (1966), 129-136.
2. , Algebraic approximation of structures over complete local rings, Inst. Hautes Etudes Sci.
Publ. Math. 36 (1969), 23-58.
3. , Algebraization of formal modules. II. Existence of modifications, Ann. of Math. (2) 91
(1970), 88-135.
4. , Algebraic construction of Brieskorn's resolutions, J. Algebra 29 (1974), 330-348.
5. H. Bass, On the ubiquity of Gorenstein rings, Math. Z. 82 (1963), 8-28.
6. A. Beauville, Foncteurs sur les anneaux artiniens. Application aux deformations verselles, Asterisque
16 (1974), 82-104.
7. J. Briancon and J. Speder, Les conditions de Whitney impliquent "ju.(*) constant", Ann. Inst.
Fourier (Grenoble) 26 (1976), 153-164.
8. E. Brieskorn, Singular elements of semi-simple algebraic groups, Proc. Internat. Congr. Math.,
Nice 1970, Vol. 2, Gauthier-Villars, Paris, 1971, pp. 279-284.
9. , Die Hierarchie der \-Modularen Singularitaten, Manuscripta Math. 27 (1979), 183-219.
10. R. Elkik, Algebrisation du module formel d'une singularity isolee, Asterisque 16 (1974), 133-144.
11. , Singularity rationnelles et deformations, Invent. Math. 47 (1978), 139-147.
12. G. Fischer, Complex analytic geometry, Lecture Notes in Math., vol. 538, Springer-Verlag, Berlin
and New York, 1976.
13. H. Grauert, Ein Theorem der analytischen Garbentheorie und die Modulrdume komplexer
Strukturen, Inst. Hautes Etudes Sci. Publ. Math. 5 (1960).
14. , Uber Modifikationen und exzeptionelle analytische Mengen, Math. Ann. 146 (1962),
15. H. Grauert and H. Kerner, Deformationen von Singularitaten komplexer Raume, Math. Ann. 153
(1964), 236-260.
16. H. Grauert and O. Riemenschneider, Verschwindungssatze fiir analytische Kohomologiegruppen
auf komplexen Raumen, Invent. Math. 11 (1970), 263-292.
17. J. Herzog and E. Kunz, Der kanonische Modul eines Cohen-Macaulay-Rings, Lecture Notes in
Math., vol. 238, Springer-Verlag, Berlin and New York, 1971.
Previous Page Next Page