The projective analytic spectrum of the double of a module

Thiago da Silva; Terence Gaffney

doi:10.14244/lajm.v5i1.96

Autores/as

Thiago da Silva Universidade Federal do Espírito Santo https://orcid.org/0000-0002-3152-0987
Terence Gaffney Northeastern University

DOI:

https://doi.org/10.14244/lajm.v5i1.96

Palabras clave:

Double of a module, Analytic projectivized spectrum, Bi-Lipschitz equisingularity

Resumen

In this work, we investigate the projectivized analytic spectrum of the double of a module. We prove that, away from the diagonal, each fiber is canonically isomorphic to the join of the corresponding fibers of the projectivized analytic spectrum of the original module. We then analyze the exceptional fiber over the origin in $C\times C$, where $C$ is an irreducible curve contained in a hypersurface, and obtain a complete description of this fiber in the case of irreducible plane curves.

Biografía del autor/a

Thiago da Silva, Universidade Federal do Espírito Santo

Thiago da Silva is an associate professor at Federal University of Espirito Santo.

Terence Gaffney, Northeastern University

Terence Gaffney is an emeritus professor at Northeastern University.

Citas

[1] A. K. Bousfield and D. M. Kan, Homotopy Limits, Completions and Localizations, Lecture Notes in Mathematics, Vol. 304, Springer-Verlag, Berlin-Heidelberg-NewYork, 1972. DOI: https://doi.org/10.1007/978-3-540-38117-4

[2] L. Birbrair, Local bi-Lipschitz classification of 2-dimensional semialgebraic sets, HoustonJ. Math. 25 (1999), no.3, 453–472.

[3] T. da Silva, Categorical Aspects of Gaffney’s double structure of a module, Mat. Contemp. 53 (2023), 213–232. DOI: https://doi.org/10.21711/231766362023/rmc5310

[4] T. da Silva and T. Gaffney, Infinitesimal Lipschitz conditions on a family of analytic varieties: genericity and necessity, São Paulo J. Math. Sci. 18 (2024), no. 2, 1207–1238. DOI: https://doi.org/10.1007/s40863-024-00452-5

[5] T. da Silva and T. Gaffney, The generic equivalence among the Lipschitz saturation of a sheaf of modules, Res. Math. Sci. 11 (2024), no. 2, Paper No. 32. DOI: https://doi.org/10.1007/s40687-024-00442-1

[6] T. da Silva, N. G. Grulha Jr., and M. Pereira, The Bi-Lipschitz Equisingularity of Essentially Isolated Determinantal Singularities, Bull. Braz. Math. Soc. (N.S.) 49 (2018), no. 4, 637–645. DOI: https://doi.org/10.1007/s00574-017-0067-3

[7] T. da Silva, N. G. Grulha Jr., and M. Pereira, Real and complex integral closure, Lipschitz equisingularity and applications on square matrices, J. Singul. 22(2020), 215–226. DOI: https://doi.org/10.5427/jsing.2020.22o

[8] A. Fernandes and M. A. S. Ruas, Bilipschitz determinacy of quasihomogeneous germs, Glasgow Math. J. 46(2004), no.1,77–82. DOI: https://doi.org/10.1017/S001708950300154X

[9] A. Fernandes and M. A. S. Ruas, Rigidity of bi-Lipschitz equivalence of weighted homogeneous function-germs in the plane, Proc. Amer. Math. Soc. 141 (2013), no. 4, 1125–1133. DOI: https://doi.org/10.1090/S0002-9939-2012-11388-7

[10] T. Gaffney, The genericity of the infinitesimal Lipschitz condition for hypersurfaces, J. Singul. 10 (2014),108–123. DOI: https://doi.org/10.5427/jsing.2014.10g

[11] T. Gaffney, Bi-Lipschitz equivalence, integral closure and invariants, Real and Complex Singularities (M. Manoel, M. C. Romero Fuster, and C. T. C. Wall, eds.), London Math. Soc. Lecture Note Ser., vol. 380, Cambridge University Press, Cambridge, 2010, pp. 112–137. DOI: https://doi.org/10.1017/CBO9780511731983.010

[12] T. Gaffney, Integral Closure of Modules and Whitney equisingularity, Invent. Math. 107 (1992), no. 2, 301–322. DOI: https://doi.org/10.1007/BF01231892

[13] T. Gaffney, Generalized Buchsbaum-Rim Multiplicities and a Theorem of Rees, Commun.Algebra 31 (2003), no. 8, 3811–3828. DOI: https://doi.org/10.1081/AGB-120022444

[14] T. Gaffney, Equisingularity of Plane Sections, t1 Condition, and the Integral Closure of Modules, Realand Complex Singularities (W.L.Marar,ed.), Pitman Res. Notes Math. Ser., vol. 333, Longman, Harlow, 1995.

[15] T. Gaffney, The Multiplicity Polar Theorem, preprint, arXiv:math/0703650 [math.CV], 2007.

[16] T. Gaffney and S. Kleiman, Specialization of integral dependence for modules, Invent. Math. 137 (1999),no. 3,541–574. DOI: https://doi.org/10.1007/s002220050335

[17] S. Kleiman and A. Thorup, A geometric theory of the Buchsbaum-Rim multiplicity, J. Algebra 167 (1994), no. 1, 168–231. DOI: https://doi.org/10.1006/jabr.1994.1182

[18] M. Lejeune-Jalabert and B. Teissier, Clôture intégrale desidéaux et equisingularité, Séminaire Lejeune-Teissier, Centre de Mathématiques École Polytechnique, 1974; Publ. Inst. Fourier, St. Martind’Hères, 1975.

[19] J. Lipman, Relative Lipschitz-saturation, Amer. J. Math. 97 (1975), no. 3, 791–813. DOI: https://doi.org/10.2307/2373777

[20] E. J. N. Looĳenga,Isolatedsingularpointsoncompleteintersections,LondonMath.Soc.Lecture

NoteSer.,vol.77,Cambridge University Press,Cambridge, 1984.

[21] T. Mostowski, A criterion for Lipschitz equisingularity, Bull. Polish Acad. Sci. Math. 37 (1989), no.1-6, 109–116 (1990).

[22] T. Mostowski, Tangent cones and Lipschitz stratifications, Singularities (Warsaw, 1985), Banach Center Publ., vol.20, PWN, Warsaw, 1988, pp. 303–322. DOI: https://doi.org/10.4064/-20-1-303-322

[23] A. Parusiński, Lipschitz stratification of real analytic sets, Singularities (Warsaw, 1985), Banach Center Publ., vol.20, PWN, Warsaw, 1988, pp. 323–333. DOI: https://doi.org/10.4064/-20-1-323-333

[24] F. Pham and B. Teissier, Fractions lipschitziennes d’une algèbre analytique complexe et saturation de Zariski, Centre de Mathématiques de l’École Polytechnique, Paris, 1969.

[25] B. Teissier, Complex curve singularities: a biased introduction, Singularities in Geometry and Topology, World Scientific Publishing, Hackensack, NJ, 2007, pp. 825–887. DOI: https://doi.org/10.1142/9789812706812_0029

[26] O. Zariski, General Theory of Saturation and of Saturated Local Rings. II. Saturated local rings of dimension 1, Amer. J. Math. 93 (1971), no. 4, 872–964. DOI: https://doi.org/10.2307/2373741