Singularities of frontal surfaces. (English) Zbl 1537.32106
Summary: We consider singularities of frontal surfaces of corank one and finite frontal codimension. We look at the classification under \(\mathscr{A}\)-equivalence and introduce the notion of frontalisation for singularities of fold type. We define the cuspidal and the transverse double point curves and prove that the frontal has finite codimension if and only if both curves are reduced. Finally, we also discuss about the frontal versions of the Marar-Mond formulas and Mond’s conjecture.
MSC:
| 32S30 | Deformations of complex singularities; vanishing cycles |
| 32S25 | Complex surface and hypersurface singularities |
Software:
SINGULARReferences:
| [1] | Arnol’d V. I., Critical points of smooth functions. In Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 1, pages 19-39, 1975. · Zbl 0343.58002 |
| [2] | Arnol’d V. I., Singularities of caustics and wave fronts, volume 62 of Math-ematics and its Applications (Soviet Series). Kluwer Academic Publishers Group, Dordrecht, 1990. · Zbl 0734.53001 |
| [3] | Buchweitz R.-O. and Greuel G.-M., The Milnor number and deformations of complex curve singularities. Invent. Math. 58 (1980), 241-281. · Zbl 0458.32014 |
| [4] | Burghelea D. and Verona A., Local homological properties of analytic sets. Manuscripta Math. 7 (1972), 55-66. · Zbl 0246.32007 |
| [5] | Chen L., Pei D. H. and Takahashi M., Dualities and envelopes of one-parameter families of frontals in hyperbolic and de Sitter 2-spaces. Math. Nachr. 293 (2020), 893-909. · Zbl 1523.53005 |
| [6] | de Jong T. and Pfister G., Local analytic geometry. Advanced Lectures in Mathematics. Friedr. Vieweg & Sohn, Braunschweig, 2000. Basic theory and applications. · Zbl 0959.32011 |
| [7] | Fujimori S., Saji K., Umehara M. and Yamada K., Singularities of maximal surfaces. Math. Z. 259 (2008), 827-848. · Zbl 1145.57026 |
| [8] | Gabrièlov A. M., Bifurcations, Dynkin diagrams and the modality of isolated singularities. Funkcional. Anal. i Priložen. 8 (1974), 7-12. |
| [9] | Giménez Conejero R. and Nuño-Ballesteros J. J., The Image Milnor Num-ber And Excellent Unfoldings. Q. J. Math. 73 (2022), 45-63. · Zbl 1487.32157 |
| [10] | Grauert H. and Remmert R., Theory of Stein spaces. Classics in Mathe-matics. Springer-Verlag, Berlin, 2004. Translated from the German by Alan Huckleberry, Reprint of the 1979 translation. · Zbl 1137.32001 |
| [11] | Hernandes M. E., Miranda A. J. and Peñafort Sanchis G., An algorithm to compute a presentation of pushforward modules. Topology Appl. 234 (2018), 440-451. · Zbl 1381.32012 |
| [12] | Ishikawa G., Infinitesimal deformations and stability of singular Legendre submanifolds. Asian J. Math. 9 (2005), 133-166. · Zbl 1086.58022 |
| [13] | Ishikawa G., Recognition problem of frontal singularities. J. Singul. 21 (2020), 149-166. · Zbl 1442.57011 |
| [14] | Lazzeri F., A theorem on the monodromy of isolated singularities. pages 269-275. Astérisque, Nos. 7 et 8, 1973. · Zbl 0301.32011 |
| [15] | Lojasiewicz S., Introduction to complex analytic geometry. Birkhäuser Ver-lag, Basel, 1991. Translated from the Polish by Maciej Klimek. · Zbl 0747.32001 |
| [16] | Marar W. L., Montaldi J. A. and Ruas M. A. S., Multiplicities of zero-schemes in quasihomogeneous corank-1 singularities C n → C n . In Singu-larity theory (Liverpool, 1996), volume 263 of London Math. Soc. Lecture Note Ser., pages 353-367. Cambridge Univ. Press, Cambridge, 1999. · Zbl 0947.32017 |
| [17] | Marar W. L., Nuño-Ballesteros J. J. and Peñafort Sanchis G., Double point curves for corank 2 map germs from C 2 to C 3 . Topology Appl. 159 (2012), 526-536. · Zbl 1245.58018 |
| [18] | Marar W. L. and Tari F., On the geometry of simple germs of co-rank 1 maps from R 3 to R 3 . Math. Proc. Cambridge Philos. Soc. 119 (1996), 469-481. · Zbl 1005.57501 |
| [19] | Marar W. L. and Mond D., Multiple point schemes for corank 1 maps. J. London Math. Soc. (2) 39 (1989), 553-567. · Zbl 0691.58015 |
| [20] | Martins R. and Nuño-Ballesteros J. J., The link of a frontal surface singu-larity. In Real and complex singularities, volume 675 of Contemp. Math., pages 181-195. Amer. Math. Soc., Providence, RI, 2016. · Zbl 1379.58017 |
| [21] | Matsumura H., Commutative ring theory, volume 8 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, second edition, 1989. Translated from the Japanese by M. Reid. · Zbl 0666.13002 |
| [22] | Milnor J., Singular points of complex hypersurfaces. Annals of Mathematics Studies, No. 61. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1968. · Zbl 0184.48405 |
| [23] | Mond D., On the classification of germs of maps from R 2 to R 3 . Proc. London Math. Soc. (3) 50 (1985), 333-369. · Zbl 0557.58006 |
| [24] | Mond D., Vanishing cycles for analytic maps. In Singularity theory and its applications, Part I (Coventry, 1988/1989), volume 1462 of Lecture Notes in Math., pages 221-234. Springer, Berlin, 1991. · Zbl 0745.32020 |
| [25] | Mond D., Some open problems in the theory of singularities of mappings. J. Singul. 12 (2015), 141-155. · Zbl 1318.32035 |
| [26] | Mond D. and Nuño-Ballesteros J. J., Singularities of mappings-the local behaviour of smooth and complex analytic mappings, volume 357 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. · Zbl 1448.58032 |
| [27] | Springer, Cham, [2020] c ⃝2020. |
| [28] | Muñoz-Cabello C., Singular libraries. https://cuspidalcoffee.github. io/libraries.html. Accessed: 2022-10-20. |
| [29] | Muñoz-Cabello C., Nuño-Ballesteros J. J. and Oset Sinha R., Deformations of corank 1 frontals. Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 2023. |
| [30] | Murata S. and Umehara M., Flat surfaces with singularities in Euclidean 3-space. J. Differential Geom. 82 (2009), 279-316. · Zbl 1184.53015 |
| [31] | Nuño-Ballesteros J. J., Unfolding plane curves with cusps and nodes. Proc. Roy. Edinburgh Sect. A 145 (2015), 161-174. · Zbl 1314.32040 |
| [32] | Oset Sinha R. and Saji K., On the geometry of folded cuspidal edges. Rev. Mat. Complut. 31 (2018), 627-650. · Zbl 1404.57049 |
| [33] | Saji K, Umehara M and Yamada K., The geometry of fronts. Ann. of Math. (2) 169 (2009), 491-529. · Zbl 1177.53014 |
| [34] | Teissier B., The hunting of invariants in the geometry of discriminants. In Real and complex singularities (Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976), pages 565-678, 1977. · Zbl 0388.32010 |
| [35] | Le D. T., Une application d’un théorème d’A’Campo à l’équisingularité. volume 35, pages 403-409, 1973. · Zbl 0271.14001 |
| [36] | Le D. T., Le concept de singularité isolée de fonction analytique. In Complex analytic singularities, volume 8 of Adv. Stud. Pure Math., pages 215-227. North-Holland, Amsterdam, 1987. · Zbl 0656.32005 |
| [37] | Zakalyukin V. M. and Kurbatskiȋ A. N., Envelope singularities of families of planes in control theory. Tr. Mat. Inst. Steklova 262(Optim. Upr.) (2008), 73-86. · Zbl 1152.93017 |
| [38] | Zariski O., Some open questions in the theory of singularities. Bull. Amer. Math. Soc. 77 (1971), 481-491. · Zbl 0236.14002 |
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