Abstract
We study a plane curve C whose singular points are cusps and the surface which is a triple covering of ℙ2 branched along C. As a result, especially we obtain an inequality for the sum of the Milnor numbers at the singularities of C and new surfaces of general type.
Similar content being viewed by others
References
BARTH, W., PETERS, C., VAN DE VEN, A.: Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete 3.Folge, Band4, Berlin-Heidelberg-New York: Springer 1983
BEAUVILLE, A.: L'application canonique pour les surfaces de type général. Inv. Math.55, 121–140(1979)
HIRZEBRUCH, F.: Some examples of algebraic surfaces. Contemporary Math.9, 55–71(1982)
KODAIRA, K.: Pluricanonical systems on algebraic surfaces of general type. J. Math. Soc. Japan20, 170–192(1968)
KODAIRA, K.: On the structure of compact analytic surfaces. IV. Amer. J. Math.90, 1048–1066(1968)
LAUFER, H.B.: Normal two-dimensional singularities, Princeton Univ. Press 1971
LAUFER, H.B.: On μ for surface singularities. Proc. Symp. in Pure Math.30, 45–49(1977)
MIRANDA, R.: Triple covers in algebraic geometry. Amer. J. Math.107, 1123–1158(1985)
MORI, S.: On a generalization of complete intersections. J. Math. Kyoto Univ.15, 619–646(1975)
PERSSON, U.: Double coverings and surfaces of general type. Lect. Notes in Math.732, 168–195(1978): Springer
YOSHIHARA, H.: A note on the existence of some curves. Algebraic Geometry and Commutative Algebra in Honor of M. Nagata. 801–804 (1987)
YOSHIHARA, H.: Plane curves whose singular points are cusps. Proc Amer. Math. Soc.103, 737–740(1988)
ZARISKI, O.: On the linear connection index of the algebraic surfaces zn=f(x,y). Proc. Nat. Acad. Sci.15, 494–501(1929)
MATSUOKA, T., SAKAI, F.: The degree of rational unicuspidal curves. (to appear)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Yoshihara, H. Plane curves whose singular points are cusps and triple coverings of ℙ2 . Manuscripta Math 64, 169–187 (1989). https://doi.org/10.1007/BF01160117
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01160117