Curves in P2(C) with 1-dimensional symmetry.

Plessis, A. A. du, and Wall, Charles Terence Clegg. "Curves in P2(C) with 1-dimensional symmetry.." Revista Matemática Complutense 12.1 (1999): 117-131. <http://eudml.org/doc/44433>.

@article{Plessis1999,
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 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.},
author = {Plessis, A. A. du, Wall, Charles Terence Clegg},
journal = {Revista Matemática Complutense},
keywords = {Espacio proyectivo complejo; Simetría; 1-dimensional symmetry group; plane curve; classification of singular points; Tjurina number},
language = {eng},
number = {1},
pages = {117-131},
title = {Curves in P2(C) with 1-dimensional symmetry.},
url = {http://eudml.org/doc/44433},
volume = {12},
year = {1999},
}

TY - JOUR
AU - Plessis, A. A. du
AU - Wall, Charles Terence Clegg
TI - Curves in P2(C) with 1-dimensional symmetry.
JO - Revista Matemática Complutense
PY - 1999
VL - 12
IS - 1
SP - 117
EP - 131
AB - 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.
LA - eng
KW - Espacio proyectivo complejo; Simetría; 1-dimensional symmetry group; plane curve; classification of singular points; Tjurina number
UR - http://eudml.org/doc/44433
ER -