Abstract
This paper applies the multiplicity polar theorem to the study of hypersurfaces with non-isolated singularities. The multiplicity polar theorem controls the multiplicity of a pair of modules in a family by relating the multiplicity at the special fiber to the multiplicity of the pair at the general fiber. It is as important to the study of multiplicities of modules as the basic theorem in ideal theory which relates the multiplicity of an ideal to the local degree of the map formed from the generators of a minimal reduction. In fact, as a corollary of the theorem, we show here that for M a submodule of finite length of a free module F over the local ring of an equidimensional complex analytic germ, that the number of points at which a generic perturbation of a minimal reduction of M is not equal to F, is the multiplicity of M.
Specifically, we apply the multiplicity polar theorem to the study of stratification conditions on families of hypersurfaces, obtaining the first set of invariants giving necessary and sufficient conditions for the Af condition for hypersurfaces with non-isolated singularities.
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References
J. Briançon, P. Maisonobe and M. Merle, Localisation de systèmes différentiels, stratifications de Whitney et condition de Thom, Invent. Math. 117 (1994), 531–50
J. Briançon and J.P. Speder, La trivialité topologique n’implique pas les conditions de Whitney, C.R. Acad. Sc.Paris, v. 280 365–367 (1975)
D.A. Buchsbaum and D.S. Rim, A generalized Koszul complex. II. Depth and multiplicity, Trans. Amer. Math. Soc. 111 (1963), 197–224
J. Fernández de Bobadilla, Approximations of non-isolated singularities of finite codimension with respect to an isolated complete intersection singularity. Bull. London Math. Soc. 35 (2003), no. 6, 812–816
W. Fulton, “Intersection Theory,” Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge · Band 2, Springer-Verlag, Berlin, 1984
T. Gaffney, Aureoles and integral closure of modules, in “Stratifications, Singularities and Differential Equations II”, Travaux en Cours 55, Herman, Paris, 1997, 55–62
T. Gaffney, Integral closure of modules and Whitney equisingularity, Invent. Math. 107 (1992), 301–22
T. Gaffney, Plane sections, Wf and Af, in “Real and complex singularities (São Carlos, 1998)”, Chapman and Hall Res. Notes Math. 412, 2000, 17–32
T. Gaffney, Multiplicities and equisingularity of ICIS germs, Invent. Math. 123 (1996), 209–220
T. Gaffney, Generalized Buchsbaum-Rim Multiplicities and a Theorem of Rees, Communications in Algebra, vol 31 #8 pp. 3811–3828, 2003
T. Gaffney, Polar methods, invariants of pairs of modules and equisingularity, Real and Complex Singularities (Sao Carlos, 2002), Ed. T. Gaffney and M. Ruas, Contemp. Math., #354, Amer. Math. Soc., Providence, RI, June 2004, 113–136
T. Gaffney, The Multiplicity-Polar Formula and Equisingularity, in preparation
T. Gaffney and R. Gassler, Segre numbers and hypersurface singularities, J. Algebraic Geom. 8 (1999), 695–736
T. Gaffney and S. Kleiman, Specialization of integral dependence for modules Invent. Math. 137 (1999), 541–574
T. Gaffney and S. Kleiman, Wf and specialization of integral dependence for modules, in “Real and complex singularities (São Carlos, 1998)”, Chapman and Hall Res. Notes Math. 412, 2000, 33–45
T. Gaffney and D. Massey, Trends in equisingularity, London Math. Soc. Lecture Note Ser. 263 (1999), 207–248
R. Gassler, Segre numbers and hypersurface singularities. Thesis, Northeastern University, 1999
J.P.G. Henry and M. Merle, Conormal Space and Jacobian module. A short dictionary, in “Proceedings of the Lille Congress of Singularities,” J.-P. Brasselet (ed.), London Math. Soc. Lecture Notes 201 (1994), 147–74
G. Jiang, Functions with non-isolated singularities on singular spaces, Thesis, Universiteit Utrecht, 1997
S. Kleiman and A. Thorup, A geometric theory of the Buchsbaum-Rim multiplicity, J. Algebra 167 (1994), 168–231
S. Kleiman and A. Thorup, The exceptional fiber of a generalized conormal space, in “New Developments in Singularity Theory.” D. Siersma, C.T.C. Wall and V. Zakalyukin (eds.), Nato Science series, II Mathematics, Physics and Chemistry, Vol. 21 2001 401–404
D.T. Lê and B. Teissier, Cycles evanescents, sections planes et conditions de Whitney. II, Proc. Sympos. Pure Math. 40, part 2, Amer. Math. Soc., 1983, 65–104
M. Lejeune-Jalabert and B. Teissier, Clôture intégrale des ideaux et equisingularité, chapitre 1 Publ. Inst. Fourier (1974)
D. Massey Lê Cycles and Hypersurface Singularities, Springer Lecture Notes in Mathematics 1615, (1995)
D. Mumford, Algebraic Geometry I Complex Projective Varieties, Springer-Verlag 1976
A. Nemethi, Hypersurface singularities with a 2-dimensional critical locus, J. London Math. Soc. (3), 59, (1999), 922–938
R. Pellikaan, Hypersurface singularities and resolutions of Jacobi modules, Thesis, Rijkuniversiteit Utrecht, 1985
R. Pellikaan, Finite determinacy of functions with non-isolated singularities, Proc. London Math. Soc. vol. 57, pp. 1–26, 1988
D. Siersma, Isolated line singularities. Singularities, Part 2 (Arcata, Calif., 1981), 485–496, Proc. Sympos. Pure Math., 40, Amer. Math. Soc., Providence, RI, 1983
B. Teissier, The hunting of invariants in the geometry of the discriminant, in “Real and complex singularities, Oslo 1976,” P. Holm (ed.), Sijthoff & Noordhoff (1977), 565–678
B. Teissier, Multiplicités polaires, sections planes, et conditions de Whitney, in “Proc. La Rábida, 1981.” J.M. Aroca, R. Buchweitz, M. Giusti and M. Merle (eds.), Springer Lecture Notes in Math. 961 (1982), 314–491
A. Zaharia, A study about singularities with non-isolated critical locus, Thesis, Rijkuniversiteit Utrecht, 1993
A. Zaharia, Topological properties of certain singularities with critical locus a 2-dimensional complete intersection, Topology Appl. 60 (1994), no. 2, 153–171
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Gaffney, T. (2006). The Multiplicity of Pairs of Modules and Hypersurface Singularities. In: Brasselet, JP., Ruas, M.A.S. (eds) Real and Complex Singularities. Trends in Mathematics. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-7776-2_11
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DOI: https://doi.org/10.1007/978-3-7643-7776-2_11
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