Plane curves with small linear orbits, I
Annales de l'Institut Fourier, Tome 50 (2000) no. 1, pp. 151-196.

L’“orbite linéaire” d’une courbe plane de degré d est son orbite dans d(d+3)/2 pour l’action naturelle de PGL (3). Dans cet article nous calculons le degré de l’adhérence de l’orbite linéaire pour la plupart des courbes dont le stabilisateur est de dimension positive. Nous utilisons une variété non singulière dominant l’adhérence de l’orbite, que nous construisons par une suite d’éclatements qui reflète la suite produisant une résolution plongée de la courbe. Les résultats obtenus ainsi seront utiles à la détermination de l’information analogue pour les courbes planes quelconques. Les orbites linéaires des courbes planes lisses ont été étudiées par les auteurs dans J. of Alg. Geom., 2 (1993), 155-184.

The “linear orbit” of a plane curve of degree d is its orbit in d(d+3)/2 under the natural action of PGL (3). In this paper we compute the degree of the closure of the linear orbits of most curves with positive dimensional stabilizers. Our tool is a nonsingular variety dominating the orbit closure, which we construct by a blow-up sequence mirroring the sequence yielding an embedded resolution of the curve. The results given here will serve as an ingredient in the computation of the analogous information for arbitrary plane curves. Linear orbits of smooth plane curves were studied by the authors in J. of Alg. Geom., 2 (1993), 155-184.

@article{AIF_2000__50_1_151_0,
     author = {Aluffi, Paoli and Faber, Carel},
     title = {Plane curves with small linear orbits, {I}},
     journal = {Annales de l'Institut Fourier},
     pages = {151--196},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {50},
     number = {1},
     year = {2000},
     doi = {10.5802/aif.1750},
     zbl = {0953.14030},
     mrnumber = {2002d:14083},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.1750/}
}
TY  - JOUR
AU  - Aluffi, Paoli
AU  - Faber, Carel
TI  - Plane curves with small linear orbits, I
JO  - Annales de l'Institut Fourier
PY  - 2000
DA  - 2000///
SP  - 151
EP  - 196
VL  - 50
IS  - 1
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.1750/
UR  - https://zbmath.org/?q=an%3A0953.14030
UR  - https://www.ams.org/mathscinet-getitem?mr=2002d:14083
UR  - https://doi.org/10.5802/aif.1750
DO  - 10.5802/aif.1750
LA  - en
ID  - AIF_2000__50_1_151_0
ER  - 
Aluffi, Paoli; Faber, Carel. Plane curves with small linear orbits, I. Annales de l'Institut Fourier, Tome 50 (2000) no. 1, pp. 151-196. doi : 10.5802/aif.1750. http://www.numdam.org/articles/10.5802/aif.1750/

[Alu] P. Aluffi, The enumerative geometry of plane cubics I: smooth cubics, Trans. AMS, 317 (1990), 501-539. | MR 90k:14058 | Zbl 0703.14035

[AF1] P. Aluffi, C. Faber, Linear orbits of smooth plane curves, J. Alg. Geom., 2 (1993), 155-184. | MR 94e:14032 | Zbl 0804.14015

[AF2] P. Aluffi, C. Faber, Linear orbits of d-tuples of points in ℙ1, J. reine & angew. Math., 445 (1993), 205-220. | MR 94j:14044 | Zbl 0781.14036

[AF3] P. Aluffi, C. Faber, A remark on the Chern class of a tensor product, Manu. Math., 88 (1995), 85-86. | MR 96e:14002 | Zbl 0863.14007

[AF4] P. Aluffi, C. Faber, Plane curves with small linear orbits II, Preprint, math.AG/9906131.

[Ful] W. Fulton, Intersection Theory, Springer Verlag, 1984. | MR 85k:14004 | Zbl 0541.14005

[Ghi] A. Ghizzetti, Sulle curve limiti di un sistema continuo ∞1 di curve piane omografiche, Memorie R. Accad. Sci. Torino (2), 68 (1937), 124-141. | JFM 62.1443.01 | Zbl 0015.26802

[MX] J.M. Miret, S. Xambó, Geometry of Complete Cuspidal Cubics, in Algebraic curves and projective geometry (Trento, 1988), Springer Lecture Notes in Math. 1389, 195-234. | Zbl 0688.14050

Cité par Sources :