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.