RMC - Vol. 12, Num. 1 - A. A. du Plessis/C. T. C. Wall

Full paper in PDF:
$%A. A. du Plessis and C. T. C. Wall, Curves in P2(C) with 1-dimensional symmetry, Rev. Mat. Complut. 12 (1999), no. 1, 117–132. %$

Curves in P₂(C) with 1-Dimensional Symmetry

Andrew A. DU PLESSIS and Charles T. C. WALL

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

Received: September 9, 1998

ABSTRACT

In a previous paper we showed that the existence of a $1$ -parameter symmetry group of a hypersurface $X$ in projective space was equivalent to failure of versality of a certain unfolding. Here we study in detail (reduced) plane curves of degree $d≥ 3$ , excluding the trivial case of cones.

We enumerate all possible group actions — these have to be either semisimple or unipotent — for any degree $d$ . A $2$ -parameter group can only occur if $d= 3$ . Explicit lists of singularities of the corresponding curves are given in the cases $d ≥ 6$ . We also show that the projective classification of these curves coincides — except in the case of the group action with weights $[-1,0,1]$ — with the classification of the singular points.

The sum $τ$ of the Tjurina numbers of the singular points is either $d2 - 3d +3$ or $d2- 3d +2$ while, for $d≥ 5$ , if there is no group action we have $τ ≤ d2- 4d+ 7$ . We give $μ= τ$ in the semi-simple case; in the unipotent case, we determine the values of both $μ$ and $τ$ .

In the semi-simple case, we show that the unfolding mentioned above is also topologically versal if $d ≥ 6$ ; in the unipotent case this holds at least if $d= 6$ .

1991 Mathematics Subject Classification: 14B05, 14NO5.