Full paper in PDF:
$%A. A. du Plessis and C. T. C. Wall, Curves in P2(C) with 1-dimensional symmetry,
Rev. Mat. Complut. 12 (1999), no. 1, 117–132. %$
Matematisk Institut ny Munkegade Aarhus Universitet 8000 Aarhus C. — Denmark
matad@mi.aau.dk | Department of Pure Mathematics The University of Liverpool Liverpool I. 69 3BX — England
C.T.C.Wall@liverpool.ac.uk |
ABSTRACT
In a previous paper we showed that the existence of a -parameter symmetry group of a hypersurface in projective space was equivalent to failure of versality of a certain unfolding. Here we study in detail (reduced) plane curves of degree , excluding the trivial case of cones.
We enumerate all possible group actions — these have to be either semisimple or unipotent — for any degree . A -parameter group can only occur if . Explicit lists of singularities of the corresponding curves are given in the cases . We also show that the projective classification of these curves coincides — except in the case of the group action with weights — with the classification of the singular points.
The sum of the Tjurina numbers of the singular points is either or while, for , if there is no group action we have . We give in the semi-simple case; in the unipotent case, we determine the values of both and .
In the semi-simple case, we show that the unfolding mentioned above is also topologically versal if ; in the unipotent case this holds at least if .
1991 Mathematics Subject Classification: 14B05, 14NO5.