Abstract
Let (X, 0) be an ICIS of dimension 2 and let \(f:(X,0)\rightarrow (\mathbb C^2,0)\) be a map germ with an isolated instability. We look at the invariants that appear when \(X_s\) is a smoothing of (X, 0) and \(f_s:X_s\rightarrow B_\epsilon \) is a stabilization of f. We find relations between these invariants and also give necessary and sufficient conditions for a 1-parameter family to be Whitney equisingular. As an application, we show that a family \((X_t,0)\) is Zariski equisingular if and only if it is Whitney equisingular and the numbers of cusps and double folds of a generic linear projection are constant with respect to t.
Similar content being viewed by others
References
Buchweitz, R.O., Greuel, G.M.: The Milnor number and deformations of complex curve singularities. Invent. Math. 58(3), 241–248 (1980)
Briançon, J., Speder, J.P.: Familles équisingulières de surfaces à singularité isolée. C. R. Acad. Sci. Paris Sér. A-B 280(Aii), A1013–A1016 (1975)
Eagon, J.A., Hochster, M.: Cohen–Macaulay rings, invariant theory, and the generic perfection of determinantal loci. Am. J. Math. 93, 1020–1058 (1971)
Gaffney, T.: Polar multiplicities and equisingularity of map germs. Topology 32(1), 185–223 (1993)
Gaffney, T., Mond, D.M.Q.: Cusps and double folds of germs of analytic maps \({\mathbb{C}}^{2}\rightarrow {\mathbb{C}}^2\). J. Lond. Math. Soc. (2) 43(1), 185–192 (1991)
Gaffney, T., Mond, D.M.Q.: Weighted homogeneous maps from the plane to the plane. Math. Proc. Camb. Philos. Soc. 109(3), 451–470 (1991)
Gibson, C.G., Wirthmuller, K., du Plessis, A.A., Looijenga, E.J.N.: Topological stability of smooth mappings. In: Lecture Notes in Mathematics, vol. 552. Springer, Berlin (1976)
Goryunov, V.: Singularities of projections of complete intersections. J. Soviet Math. 27, 2785–2811 (1984)
Greuel, G.M.: Dualitat in der lokalen kohomologie isolierter singularitaten. Math Ann. 250, 157–173 (1980)
Henry, J.P., Merle, M.: Fronces et doubles plis. Compos. Math. 101(1), 21–54 (1996)
Hernandes, M.E., Miranda, A.J., Peñafort-Sanchis, G.: An algorithm to compute a presentation of pushforward modules. Preprint arXiv:1703.03357 [math.AG]
Marar, W.L., Nuño-Ballesteros, J.J., Peñafort-Sanchis, G.: Double point curves for corank 2 map germs from \({\mathbb{C}}^{2}\) to \({\mathbb{C}}^{3}\). Topol. Appl. 159(2), 526–536 (2012)
Matsumura, H.: Commutative Ring Theory, 2nd Edition. Cambridge Studies in Advanced Mathematics, vol. 8. Cambridge University Press, Cambridge (1989)
Milnor, J., Orlik, P.: Isolated singularities defined by weighted homogeneous polynomials. Topology 9, 385–393 (1970)
Mond, D., Montaldi, J.: Deformations of maps on complete intersections. In: Damon’s \({\fancyscript {K}}_{V}\) Equivalence and Bifurcations. London Mathematical Society Lecture Note 201, pp 262–284. Cambridge University Press (1994)
Mond, D., Pellikaan, R.: Fitting ideals and multiple points of analytic mappings. In: Algebraic Geometry and Complex Analysis (Pátzcuaro, 1987), Lecture Notes in Mathematical, vol. 1414, pp. 107–161. Springer, Berlin (1989)
Nuño-Ballesteros, J.J., Oréfice-Okamoto, B., Tomazella, J.N.: Non-negative deformations of weighted homogeneous singularities. Glasg. Math. J. (To appear)
Nuño-Ballesteros, J.J., Oréfice-Okamoto, B., Tomazella, J.N.: Equisingularity of families of isolated determinantal singularities. Preprint arXiv:1405.3403 [math.AG]
Nuño-Ballesteros, J.J., Tomazella, J.N.: Equisingularity of families of map germs between curves. Math. Z. 272(1–2), 349–360 (2012)
Rieger, J.H.: Families of maps from the plane to the plane. J. Lond. Math. Soc. (2) 36(2), 351–369 (1987)
Speder, J.P.: Équisingularité et conditions de Whitney. Am. J. Math. 97(3), 571–588 (1975)
Teissier, B.: The hunting of invariants in the geometry of discriminants. Real and Complex Singularities, Oslo, pp. 565–678 (1977). Alphen aan den Rijn, Sijthoff and Noordhoff (1976)
Wall, C.T.C.: Finite determinacy of smooth map-germs. Bull. Lond. Math. Soc. 13(6), 481–539 (1981)
Whitney, H.: On singularities of mappings of euclidean spaces. I. Mappings of the plane into the plane. Ann. Math. (2) 62, 374–410 (1955)
Zariski, O.: Some open questions in the theory of singularities. Bull. Am. Math. Soc. 77, 481–491 (1971)
Author information
Authors and Affiliations
Corresponding author
Additional information
J. J. Nuño-Ballesteros is partially supported by DGICYT Grant MTM2015-64013-P and CAPES-PVE Grant 88881.062217/2014-01. B. Oréfice-Okamoto is partially supported by FAPESP Grant 2013/14014-3. J. N. Tomazella is partially supported by CNPq Grant 309626/2014-5 and FAPESP Grant 2016/04740-7.
Rights and permissions
About this article
Cite this article
Nuño-Ballesteros, J.J., Oréfice-Okamoto, B. & Tomazella, J.N. Equisingularity of map germs from a surface to the plane. Collect. Math. 69, 65–81 (2018). https://doi.org/10.1007/s13348-017-0194-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13348-017-0194-6