Abstract
We assume V a hypersurface of degree d in \({P^n({\mathbb C})}\) with isolated singularities and not a cone, admitting a group G of linear symmetries. In earlier work we treated the case when G is semi-simple; here we analyse the unipotent case. Our first main result lists the possible groups G. In each case we discuss the geometry of the action, reduce V to a normal form, find the singular points, study their nature, and calculate the Milnor numbers. The Tjurina number τ(V) ≤ (d − 1)n–2(d 2 − 3d + 3): we call V oversymmetric if this value is attained. We calculate τ in many cases, and characterise the oversymmetric situations. In particular, we list all the cases with dim(G) = 2 which are the oversymmetric cases with d = 3.
Similar content being viewed by others
References
Dimca A.: Singularities and Topology of Hypersurfaces. Springer-Verlag, New York (1992)
du Plessis, A.A., Wall, C.T.C.: Hypersurfaces with isolated singularities with symmetry. In: Saia, J.M., Seade, J. (eds.) Real and Complex Singularities. (Proceedings of the IX International Workshop). Contemp. Math. Amer. Math. Soc. 459, pp 147–164 (2008)
du Plessis A.A., Wall C.T.C.: Curves in \({P^2(\mathbb{C})}\) with 1-dimensional symmetry. Rev. Math. Complut. 12, 117–132 (1999)
du Plessis, A.A., Wall, C.T.C.: Applications of discriminant matrices, in Aspects des Singularités. In: Proceedings of Lille Singularities Semester. http://www-gat.univ-lille1.fr~tibar/Aspects/index.htm (2000)
du Plessis A.A., Wall C.T.C.: Hypersurfaces in P n with 1-parameter symmetry groups. Proc. Roy Soc. Lond. A 456, 2515–2541 (2000)
du Plessis, A.A., Wall, C.T.C.: Discriminants, vector fields and singular hypersurfaces. In: Siersma, D., Wall, C.T.C., Zakalyukin, V. (eds.) New Developments in Singularity Theory, pp. 351–377. Kluwer, The Netherlands (2001)
Nowicki A.: Polynomial Derivations and Their Rings of Constants. Uniwersytet Nikolaja Kopernika, Turun (1994)
Saito K.: Quasihomogene isolierte Singularitäten von Hyperflächen. Invent. Math. 14, 123–142 (1971)
Steenbrink J.H.M.: Intersection form for quasi-homogeneous singularities. Compos. Math. 34, 211–223 (1977)
Tan L.: An algorithm for explicit generators of the invariants of the basic G a -actions. Commun. Algebra 17, 565–572 (1989)
Wall C.T.C.: Notes on the classification of singularities. Proc. Lond. Math. Soc. 48, 461–513 (1984)
Weitzenbock R.: Ueber die invarianten von linearen Gruppen. Acta Math. 58, 231–293 (1932)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
du Plessis, A.A., Wall, C.T.C. Hypersurfaces in P n with 1-parameter symmetry groups II. manuscripta math. 131, 111–143 (2010). https://doi.org/10.1007/s00229-009-0304-1
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00229-009-0304-1