Abstract
This paper deals with the local study of invariants of analytic curve singularities. For monomial curves we obtain a numerical description of the \({\mathcal{A}}_e\) -codimension of parametrized curves in terms of classical invariants of the theory of curves, like the delta invariant, the Tjurina number and the Cohen-Macaulay type of its local ring.
Similar content being viewed by others
References
Buchweitz R.-O. (1980). On deformations of monomial curves. LNM 777: 205–220
Buchweitz R.-O. and Greuel G.-M. (1980). The Milnor number and deformations of complex curve singularities. Invent. Math. 58: 241–281
Cavaliere M.P. and Niesi G. (1983). On monomial curves and Cohen-Macaulay type. Manuscr. Math. 42: 147–159
Cavaliere, M.P., Niesi, G.: On form ring of a one-dimensional semigroup ring. In: Comm. Algebra. Proc. Trento Conf. NY. Lecture Notes in Pure and Applied Math. vol.~84, pp.~39–48. Marcel Dekker, New York (1983)
Damon J. and Mond D. (1991).\({\mathcal{A}}\) -codimension and the vanishing topology of discriminants Invent. Math. 106: 217–242
Gibson C.G. and Hobbs C.A. (1993). Simple singularities of space curves. Math. Proc. Camb. Phil. Soc. 113(2): 297–310
Greuel G.-M. (1981). On deformation of curves and a formula of Deligne. Proc. La Rabida. LNM 961: 141–168
Greuel G.-M., Lossen C. and Shustin E. (2006). Introduction to Singularities and Deformations. Springer, Berlin
Martinet, J.: Singularités des fonctions et applications différentiables. (PUC, Rio de Janeiro 1974) transl. Singularities of smooth functions and mappings. LNM 58 (1982)
Mond D. (1995). Looking at bent wires—\({\mathcal{A}}_e\) -codimension and the vanishing topology of parametrized curve singularitiesMath. Proc. Camb. Phil. Soc. 117: 213–222
Mond, D.: Vanishing cycles for analytic maps. Singularity Theory and Applications. Warwick 1989, LNM, vol. 1462, pp. 221–234. Springer, Heidelberg (1991)
Serre J.P. (1960). Sur les modules projectifs. Sem. Dubreil-Pisot 2: 1–16
Tjurina G.N. (1969). Locally semiuniversal flat deformations of isolated singularities of complex spaces. Math. USSR-Izv. 3: 967–999
Wall C.T.C. (1981). Finite determinacy of smooth map-germs. Bull. Lond. Math. Soc. 13(6): 481–539
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hernandes, M.E., Hernandes, M.E.R. & Ruas, M.A.S. \({\mathcal{A}}_e\) -codimension of germs of analytic curves. manuscripta math. 124, 237–246 (2007). https://doi.org/10.1007/s00229-007-0116-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00229-007-0116-0