Topological triviality of versal unfoldings of complete intersections

James Damon

Annales de l'institut Fourier (1984)

  • Volume: 34, Issue: 4, page 225-251
  • ISSN: 0373-0956

Abstract

top
We obtain algebraic and geometric conditions for the topological triviality of versal unfoldings of weighted homogeneous complete intersections along subspaces corresponding to deformations of maximal weight. These results are applied: to infinite families of surface singularities in C 4 which begin with the exceptional unimodular singularities, to the intersection of pairs of generic quadrics, and to certain curve singularities.The algebraic conditions are related to the operation of adjoining powers, a generalization for complete intersections of a special form of the Thom-Sebastiani operation. A duality result is proven which relates the Jacobian algebra of f being Gorenstein with N ˜ ( F ) * being principal, i.e. generated by one element (here F is obtained from f by adjoining powers, and N ˜ ( F ) * is the dual of the space of non-trivial infinitesimal deformations.

How to cite

top

Damon, James. "Topological triviality of versal unfoldings of complete intersections." Annales de l'institut Fourier 34.4 (1984): 225-251. <http://eudml.org/doc/74657>.

@article{Damon1984,
abstract = {We obtain algebraic and geometric conditions for the topological triviality of versal unfoldings of weighted homogeneous complete intersections along subspaces corresponding to deformations of maximal weight. These results are applied: to infinite families of surface singularities in $C^4$ which begin with the exceptional unimodular singularities, to the intersection of pairs of generic quadrics, and to certain curve singularities.The algebraic conditions are related to the operation of adjoining powers, a generalization for complete intersections of a special form of the Thom-Sebastiani operation. A duality result is proven which relates the Jacobian algebra of $f$ being Gorenstein with $\widetilde\{N\}(F)^*$ being principal, i.e. generated by one element (here $F$ is obtained from $f$ by adjoining powers, and $\widetilde\{N\}(F)^*$ is the dual of the space of non-trivial infinitesimal deformations.},
author = {Damon, James},
journal = {Annales de l'institut Fourier},
keywords = {exceptional unimodular singularities; Thom-Sebastiani operation; Jacobian algebra},
language = {eng},
number = {4},
pages = {225-251},
publisher = {Association des Annales de l'Institut Fourier},
title = {Topological triviality of versal unfoldings of complete intersections},
url = {http://eudml.org/doc/74657},
volume = {34},
year = {1984},
}

TY - JOUR
AU - Damon, James
TI - Topological triviality of versal unfoldings of complete intersections
JO - Annales de l'institut Fourier
PY - 1984
PB - Association des Annales de l'Institut Fourier
VL - 34
IS - 4
SP - 225
EP - 251
AB - We obtain algebraic and geometric conditions for the topological triviality of versal unfoldings of weighted homogeneous complete intersections along subspaces corresponding to deformations of maximal weight. These results are applied: to infinite families of surface singularities in $C^4$ which begin with the exceptional unimodular singularities, to the intersection of pairs of generic quadrics, and to certain curve singularities.The algebraic conditions are related to the operation of adjoining powers, a generalization for complete intersections of a special form of the Thom-Sebastiani operation. A duality result is proven which relates the Jacobian algebra of $f$ being Gorenstein with $\widetilde{N}(F)^*$ being principal, i.e. generated by one element (here $F$ is obtained from $f$ by adjoining powers, and $\widetilde{N}(F)^*$ is the dual of the space of non-trivial infinitesimal deformations.
LA - eng
KW - exceptional unimodular singularities; Thom-Sebastiani operation; Jacobian algebra
UR - http://eudml.org/doc/74657
ER -

References

top
  1. [1] V. I. ARNOLD, Local Normal Forms of Functions, Invent. Math., 35 (1976), 87-109. Zbl0336.57022MR57 #7646
  2. [2] J. W. BRUCE, A Stratification of the Space of Cubic Surfaces, Math. Proc. Camb. Phil. Soc., 87 (1980), 427-441. Zbl0434.58009MR81c:14022
  3. [3] J. W. BRUCE and P. J. GIBLIN, A Stratification of the Space of Plane Quartic Curves, Proc. London Math. Soc., (3) 42 (1981), 270-298. Zbl0403.14004MR82j:14022
  4. [3a] D. BUCHBAUM and D. EISENBUD, Algebraic structure for finite free resolutions and some structure theorems for ideals of codimension 3, Amer, J. Math., 99 (1977), 447-485. Zbl0373.13006
  5. [4] J. DAMON, Finite Determinacy and Topological Triviality I., Invent. Math., 62 (1980), 299-324. II. Sufficient Conditions and Topological Stability, Compositio Math., 47 (1982), 101-132. Zbl0489.58003MR82f:58018
  6. [5] J. DAMON, Classification of Discrete Algebra Types, preprint. 
  7. [6] J. DAMON, Topological Properties of Real Simple Germs, Curves and the Nice Dimensions n &gt; p, Math. Proc. Camb. Phil. Soc., 89 (1981), 457-472. Zbl0516.58013MR82g:58014
  8. [7] J. DAMON, Topological Properties of Discrete Algebra Type II : Real and Complex Algebras, Amer. Jour. Math., Vol. 101 No. 6 (1979), 1219-1248. Zbl0498.58005MR80k:58020
  9. [8] J. DAMON and A. GALLIGO, On the Hilbert-Samuel Partition for Stable Map-Germs to appear, Bull. Soc. Math. France. Zbl0547.58001
  10. [9] I. V. DOLGACHEV, Quotient Conical Singularities on Complex Surfaces, Funct. Anal. and Appl., 9 No. 2 (1975), 160-161. Zbl0295.14017
  11. [10] I. V. DOLGACHEV, Automorphic Forms and Quasihomogeneous Singularities, Funct. Anal. and Appl., 9 No. 2 (1975), 149-150. Zbl0321.14003MR58 #27958
  12. [11] M. GIUSTI, Classification des singularités isolées d'intersections complètes simples, C.R.A.S., Paris, t284 (17 Jan. 1977), 167-169. Zbl0346.32015MR55 #12948
  13. [12] G. M. GREUEL, Dualität in der lokalen Kohomologie Isolierter Singularitäten, Math. Ann., 250 (1980), 157-173. Zbl0417.14003MR82e:32009
  14. [13] G. M. GREUEL, Der Gauss-Manin Zusammenhang isolierter Singularitäten von vollständiger Durchschnitten, Math. Ann., 214 (1975), 235-266. Zbl0285.14002MR53 #417
  15. [14] G. M. GREUEL, Die Zahl der Spitzen und Die Jacobi-Algebra einer isolierten Hyperflächensingularität, Manuscripta Math., 21 (1977), 227-241. Zbl0359.32008MR57 #3117
  16. [15] G. M. GREUEL, and H. A. HAMM, Invarianten quasihomogener vollständiger Durchschnitte, Invent. Math., 49 (1978), 67-86. Zbl0394.32006MR80d:14003
  17. [16] H. KNÖRRER, Isolierte Singularitäten von Durchschitten Zweier Quadriken, thesis Bonn 1978, Bonner Mathematische Schriften 116. 
  18. [17] H. KNÖRRER, Die Singularitäten vom typ D, Math. Ann., 251 (1980), 135-150. Zbl0423.32011
  19. [18] E. LOOIJENGA, Semi-universal Deformation of a Simple Elliptic Hypersurface Singularity I : Unimodularity, Topology, 16 (1977), 257-262. Zbl0373.32004MR56 #8565
  20. [19] J. MATHER, Stability of C∞-Mappings. III. Finitely Determined Map Germs, Publ. Math. I.H.E.S., 35 (1968), 127-146. IV. Classification of Stable Germs by R-algebras, Publ. Math. I.H.E.S., 37 (1979), 234-248. V. Transversality, Adv. in Math., 4 (1970), 301-336. VI. The Nice Dimensions Liverpool Singularities Symposium I, Springer Lecture Notes, 192 (1970), 207-253. Zbl0211.56105
  21. [20] J. MATHER, Stratification and Mappings, Dynamical Systems, M. Peixoto Ed., Academic Press (1973), 195-232. Zbl0286.58003MR51 #4306
  22. [21] J. MATHER, How to Stratify mappings and Jet Spaces, in : Singularités d'Applications Différentiables, Plans sur Bex-1975, Springer Lecture Notes 535, 128-176. Zbl0398.58008MR56 #13259
  23. [22] J. MATHER and J. DAMON, Book on singularities of mappings, in preparation. 
  24. [23] H. C. PINKHAM, Groupes de monodromie des singularities unimodulaires exceptionnelles, C.R.A.S., Paris, t. 284 (1977), 1515-1518. Zbl0391.14005MR55 #12722
  25. [24] F. RONGA, Une Application Topologiquement Stable qui ne peut pas être approchée par une Application Différentiablement Stable, C.R.A.S., Paris, t. 287 (30 Oct. 1978), 779-782. Zbl0397.58009MR80f:58014
  26. [25] K. SAITO, Einfach-elliptische Singularitäten, Invent, Math., 23 (1974), 289-325. Zbl0296.14019MR50 #7147
  27. [26] R. THOM, Ensembles et Morphismes Stratifiés, Bull. Amer. Math. Soc., 75 (1969), 240-284. Zbl0197.20502MR39 #970
  28. [27] R. THOM and M. SEBASTIANI, Un résultat sur la monodromie, Invent. Math., 13 (1971), 90-96. Zbl0233.32025MR45 #2201
  29. [28] C. T. C. WALL, First Canonical Stratum, Jour. Lond. Math. Soc., Vol. 21 Pt 3 (1980), 419-433. Zbl0467.58008MR81k:58014
  30. [29] K. WIRTHMULLER, Universell Topologische Triviale Deformationen, Thesis, Univ. of Regensberg. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.