Resumen de Curves in P2(C) with 1-dimensional symmetry

A.A. Du Plessis, Charles Terence Clegg Wall

• 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 d³3, 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 d³5, 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 d³6; in the unipotent case this holds at least if d=6.