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 t 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 t ≤ d2 - 4d + 7. We give m = t in the semi-simple case; in the unipotent case, we determine the values of both m and t. 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.
@article{urn:eudml:doc:44433,
title = {Curves in P2(C) with 1-dimensional symmetry.},
journal = {Revista Matem\'atica de la Universidad Complutense de Madrid},
volume = {12},
year = {1999},
pages = {117-131},
zbl = {0969.14022},
mrnumber = {MR1698902},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:44433}
}
Plessis, A. A. du; Wall, Charles Terence Clegg. Curves in P2(C) with 1-dimensional symmetry.. Revista Matemática de la Universidad Complutense de Madrid, Tome 12 (1999) pp. 117-131. http://gdmltest.u-ga.fr/item/urn:eudml:doc:44433/