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

A. A. du Plessis; Charles Terence Clegg Wall

Revista Matemática Complutense (1999)

- Volume: 12, Issue: 1, page 117-131
- ISSN: 1139-1138

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topPlessis, 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 -

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