Geodesic
Multiplicity of Boardman strata and deformations of map germs
Glasgow mathematical journal, Tome 40 (1998) no. 1, pp. 21-32

Voir la notice de l'article provenant de la source Cambridge University Press

We define algebraically for each map germ f:Kn,0→Kp, 0 and for each Boardman symbol i=(i1,...,ik) a number ci(f) which is -invariant. If f is finitely determined, this number is the generalization of the Milnor number of f when p = 1, the number of cusps of f when n = p = 2, or the number of cross caps when n = 2, p = 3. We study some properties of this number and prove that, in some particular cases, this number can be interpreted geometrically as the number of Σi points that appear in a generic deformation of f. In the last part, we compute this number in the case that the map germ is a projection and give some applications to catastrophe map germs. 
Ballesteros, J. J. Nuño; Saia, M. J. Multiplicity of Boardman strata and deformations of map germs. Glasgow mathematical journal, Tome 40 (1998) no. 1, pp. 21-32. doi: 10.1017/S0017089500032328
@article{10_1017_S0017089500032328,
     author = {Ballesteros, J. J. Nu\~no and Saia, M. J.},
     title = {Multiplicity of {Boardman} strata and deformations of map germs},
     journal = {Glasgow mathematical journal},
     pages = {21--32},
     year = {1998},
     volume = {40},
     number = {1},
     doi = {10.1017/S0017089500032328},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089500032328/}
}
TY  - JOUR
AU  - Ballesteros, J. J. Nuño
AU  - Saia, M. J.
TI  - Multiplicity of Boardman strata and deformations of map germs
JO  - Glasgow mathematical journal
PY  - 1998
SP  - 21
EP  - 32
VL  - 40
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.1017/S0017089500032328/
DO  - 10.1017/S0017089500032328
ID  - 10_1017_S0017089500032328
ER  - 
%0 Journal Article
%A Ballesteros, J. J. Nuño
%A Saia, M. J.
%T Multiplicity of Boardman strata and deformations of map germs
%J Glasgow mathematical journal
%D 1998
%P 21-32
%V 40
%N 1
%U http://geodesic.mathdoc.fr/articles/10.1017/S0017089500032328/
%R 10.1017/S0017089500032328
%F 10_1017_S0017089500032328

[1] 1.Boardman, J., Singularities of differentiable maps, Inst. Hautes Études Sci. Publ. Math. 33 (1967), 21–57. Google Scholar | DOI

[2] 2.Fukuda, T. and Ishikawa, G., On the number ofcusps of stable perturbations of a plane-to-plane singularity, Tokyo J. Math., 10 (1987), 375–384. Google Scholar | DOI

[3] 3.Fulton, W., Intersection theory (Springer-Verlag, 1984). Google Scholar | DOI

[4] 4.Guffney, T. and Mond, D., Cusps and double folds of germs of analytic maps ℂ2→ℂ2, J. London Math. Soc. 2 43 (1991), 185–192. Google Scholar | DOI

[5] 5.Mather, J. N., Stable map-germs and algebraic geometry, in Manifolds-Amsterdam, Lect. Notes in Math. 197 (Springer-Verlag, 1971) 176–193. Google Scholar

[6] 6.Matsumura, H., Commutative ring theory (Cambridge Studies in Advanced Math. 8, Cambridge University Press, 1986). Google Scholar

[7] 7.Mond, D., Vanishing cycles for analytic maps, in Singularity Theory and its Applications, Lecture Notes in Math. 1462 (Springer-Verlag, 1991) 221–234. Google Scholar | DOI

[8] 8.Morin, B., Formes canoniques des singularités d'une application différentiate, C. R. Acad. Sci. Paris 260 (1965), 5662–5665, 6503–6506. Google Scholar

[9] 9.Morin, B., Calcul jacobien, Ann. Sci. Ecole Norm. Sup. 8 (1975), 1–98. Google Scholar | DOI

[10] 10.Mumford, D., Algebraic Geometry I Complex Projective Varieties, A series of Comprehensive Studies in Mathematics 221 (Springer-Verlag, Berlin, Heidelberg, 1976). Google Scholar

[11] 11.Trotman, D. J. A. and Zeeman, E. C., The classification of elementary catastrophes of codimension ≤5, in Structural Stability, the Theory of Catastrophes and Applications in the Sciences, Seattle 1975, Lecture Notes in Math. 525 (Springer-Verlag, 1976). Google Scholar

[12] 12.Wall, C. T. C., Finite determinacy of smooth map-germs, Bull. London Math. Soc. 13 (1981), 481–539. Google Scholar | DOI

Cité par Sources :