no. 1
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 ℙ 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 ℙ 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.

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     title = {Plane curves with small linear orbits, {I}},
     journal = {Annales de l'Institut Fourier},
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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 | Zbl

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

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

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

[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 | Zbl

[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 | Zbl

[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

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