Abstract
Globally irreducible nodes (i.e. nodes whose branches belong to the same irreducible component) have mild effects on the most common topological invariants of an algebraic curve. In other words, adding a globally irreducible node (simple nodal degeneration) to a curve should not change them a lot. In this paper we study the effect of nodal degeneration of curves on fundamental groups and show examples where simple nodal degenerations produce non-isomorphic fundamental groups and this can be detected in an algebraic way by means of Galois covers.
Similar content being viewed by others
References
Artal E., Carmona J., Cogolludo J.I. (2003) Braid monodromy and topology of plane curves. Duke Math. J. 118(2): 261–278
Artal, E., Cogolludo, J.I., Tokunaga, H.: Pencils and Infinite Dihedral covers of \(\mathbb{P}^{2}\). To appear in Proceedings of the American Mathematical Society. Preprint available at arXiv:math.AG/0411506.
Bannai, S.: Construction of versal G-covers via toric varieties. Submitted to Osaka J. Math. (2005)
Carmona, J.: Monodromía de Trenzas de Curvas Algebraicas Planas. Ph.D. thesis, Universidad de Zaragoza (2003)
Deligne, P.: Le groupe fondamental du complément d’une courbe plane n’ayant que des points doubles ordinaires est abélien (d’après W. Fulton), Bourbaki Seminar, vol. 1979/80, Lecture Notes in Math., vol. 842. Springer, Berlin (1981) pp. 1–10.
Dimca A. (1992) Singularities and Topology of Hypersurfaces. Universitext, Springer-Verlag, New York
Esnault H. (1982) Fibre de Milnor d’un cône sur une courbe plane singulière. Invent. Math. 68(3): 477–496
Fulton W. (1980) On the fundamental group of the complement of a node curve. Ann. Math. (2) 111(2): 407–409
Harris J. (1986) On the Severi problem. Invent. Math. 84(3): 445–461
Horikawa E. (1975) On deformation of quintic surfaces. Invent. Math. 31, 43–85
Libgober A. (1982) Alexander polynomial of plane algebraic curves and cyclic multiple planes. Duke Math. J. 49(4): 833–851
Libgober, A.: Characteristic varieties of algebraic curves. Applications of algebraic geometry to coding theory, physics and computation (Eilat, 2001) Kluwer Acad. Publ., Dordrecht, 2001, 215–254
Namba M. (1987) Branched coverings and algebraic functions Pitman Research Notes in Mathematics Series, vol 161. Longman Scientific & Technical, Harlow
Oka M., Pho D.T. Fundamental group of sextics of torus type. Trends in singularities, pp. 151–180, Trends Math. Birkhäuser, Basel (2002)
Severi F. (1921) Vorlesungen Über Algebraische Geometrie. Teubner, Leipzig
Shustin, E.: Smoothness and irreducibility of families of plane algebraic curves with ordinary singularities. Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry (Ramat Gan, 1993), pp. 393–416, Isr. Math. Conf. Proc. 9, Bar-Ilan Univ., Ramat Gan (1996)
Tokunaga H. (1994) On dihedral Galois coverings. Can. J. Math. 46: 1299–1317
Tokunaga H. (2002) Galois covers for S 4 and A 4 and their applications. Osaka. J. Math. 39, 621–645
Zariski O. (1929) On the problem of existence of algebraic functions of two variables possessing a given branch curve. Am. J. Math. 51, 305–328
Zariski O. (1931) On the irregularity of cyclic multiple planes. Ann. Math. 32, 445–489
Zariski O. (1937) The topological discriminant group of a Riemann surface of genus p. Am. J. Math. 59, 335–358
Zariski O. (1958) On the purity of the branch locus of algebraic functions. Proc. Nat. Acad. USA 44, 791–796
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bartolo, E.A., Cogolludo, J.I. & Tokunaga, Ho. Nodal degenerations of plane curves and galois covers. Geom Dedicata 121, 129–142 (2006). https://doi.org/10.1007/s10711-006-9094-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10711-006-9094-8