×

zbMATH — the first resource for mathematics

Equianalytic and equisingular families of curves on surfaces. (English) Zbl 0874.14021
The authors consider flat families of embedded reduced curves on a smooth surface \(S\) such that for each member \(C\) of the family the number of singular points of \(C\) and for each singular point \(x\in C\) the analytic (resp. the equisingular) type of \((C,x)\) is fixed, then they obtain a locally closed subscheme \(H_S^{ea}\) (resp. \(H_S^{es}\)) of the Hilbert scheme \(H_S\) of \(S\). They study local properties of \(H_S^{ea}\) (resp. \(H_S^{es})\) and give conditions on the smoothness of these subschemes. They improve previous results for curves in \(\mathbb{P}^2\) and give answer to some existing problem of curves of given degree having a fixed number of singularities of given analytic type.

MSC:
14J17 Singularities of surfaces or higher-dimensional varieties
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
14H20 Singularities of curves, local rings
32S15 Equisingularity (topological and analytic)
PDF BibTeX XML Cite
Full Text: DOI EuDML arXiv
References:
[1] [AGV] Arnold’d, V.I.; Gusein-Zade, S.M.; Varchenko, A.N.: Singularities of differentiable maps, Vol. I. Birkhäuser-Verlag (1985).
[2] [Ar] Artin, M.: Deformations of singularities. Tata Inst. Fund. Res. Lect. Notes, Vol. 54 (1976). · Zbl 0395.14003
[3] [Bin] Bingener, J.: Darstellbarkeitskrieterien für analytische Funktoren. Ann. Sci. École Norm. Sup.13, 317–347 (1980). · Zbl 0454.32017
[4] [BrK] Brieskorn, E.; Knörrer, H.: Plane algebraic curves. Birkhäuser-Verlag (1986).
[5] [De] Delgado, de la Mata, F.: The semigroup of values of a curve singularity with several branches. Manuscripta math.59, 347–374 (1987). · Zbl 0611.14025
[6] [DH] Diaz, S.; Harris, J.: Ideals associated to deformations of singular plane curves. Trans. Amer. Math. Soc.,309, No. 2, 433–467 (1988). · Zbl 0707.14022
[7] [F1] Flenner, H.: Ein Kriterium für die Offenheit der Versalität. Math. Z.178, 449–473 (1981). · Zbl 0462.14003
[8] [Gab] Gabrièlov, A.M.: Bifurcations, Dynkin-diagrams and modality of isolated singularities. Functional analysis and its appl.8, 94–98 (1974). · Zbl 0344.32007
[9] [Gia] Giacinti-Diebolt, C.: Variétés des courbes projectives planes de degré et lieu singulier donnés. Math. Ann.266, 321–350 (1984). · Zbl 0527.14024
[10] [Gr] Greuel, G.-M.: Easy deformations of curve singularities. Manuscript, University of Kaiserslautern, 1994.
[11] [GrK] Greuel G.-M.; Karras, U.: Families of varieties with prescribed singularities. Comp. Math.69, 83–110 (1989). · Zbl 0684.32015
[12] [He] Hesse, M.O.: Über die Bedingung, unter welcher eine homogene ganze Function vonn unabhängigen Variabeln durch lineäre Substitutionen vonn andern unabhängigen Variabeln auf eine homogene Function sich zurückführen läßt, die eine Variable weniger enthält. Journal für reine und angew. Math.42, 117–124 (1851). · ERAM 042.1147cj
[13] [La] Laudal, A.: Formal moduli of algebraic structures. SLN 754, Springer-Verlag (1979). · Zbl 0438.14007
[14] [Lo] Lossen, C.: Äquisinguläre Familien von Kurven auf Flächen. Diplomarbeit Univ. Kaiserslautern (1994).
[15] [Mu] Mumford, D: Lectures on curves on an algebraic surface. Princeton Univ. Press (1966). · Zbl 0187.42701
[16] [Sch] Schlessinger, M.: Functors of Artin rings. Trans. Amer. Math. Soc.130, 208–222 (1968). · Zbl 0167.49503
[17] [Sev] Severi, F.: Vorlesungen über algebraische Geometrie. Teubner (1921) resp. Johnson (1968). · JFM 48.0687.01
[18] [Sh1] Shustin, E.: Versal deformation in the space of plane curves of fixed degree. Function. Anal. Appl.21, 82–84 (1987). · Zbl 0627.14023
[19] [Sh2] Shustin, E.: On manifolds of singular algebraic curves. Sel. Math. Sov.10, No. 1, 27–37 (1991).
[20] [Sh3] Shustin, E.: Embeddings of three-cuspidal singularity in the space of curves of degree 7, Preprint.
[21] [Sh4] Shustin, E.: Geometry of equisingular families of plane algebraic curves. J. Algebr. Geom.5, 209–234 (1996). · Zbl 0861.14020
[22] [Te] Teissier, B.: Résolution simultanée I, II. In: Seminaire Demazure-Pinkham-Teissier 1976/1977, SLN 777, Springer-Verlag (1980).
[23] [Wa1] Wahl, J.M.: Deformations of plane curves with nodes and cusps. Ann. Journ. of Math.96, 529–577 (1974). · Zbl 0299.14008
[24] [Wa2] Wahl, J.M.: Equisingular deformations of plane algebroid curves. Trans. Amer. Soc.193, 143–170 (1974). · Zbl 0294.14007
[25] [Zar] Zariski, O.: Studies in equisingularity I–III. Amer. Journ. Math.87, 507–536 and 972–1006 (1965), resp. Amer. Journ. Math.90, 961–1023, (1968). · Zbl 0132.41601
[26] [Yos] Yoshihara, H.: On plane rational curves. Proc. Japan Acad.55, Ser. A, 152–155 (1979). · Zbl 0432.14019
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.