Aluffi, Paolo; Faber, Carel Plane curves with small linear orbits. I. (English) Zbl 0953.14030 Ann. Inst. Fourier 50, No. 1, 151-196 (2000). Let \(C \subset \mathbb{P}^2\) be a reduced plane curve. Here the authors classify all such \(C\) which are not union of lines and such that their orbit for the natural action of PGL(3) has dimension \(\leq 7\). This work is significant for the enumerative geometry of plane curves and in particular for the computation of the degree of the orbit closure of an arbitrary plane curve. Fur further works by the authors, see P. Aluffi and C. Faber, “Plane curves with small linear orbits. II”, e-prints math.AG/9906131 and “Linear orbits of arbitrary plane plane curves”, math.AG/9912092. Reviewer: E.Ballico (Povo) Cited in 2 ReviewsCited in 5 Documents MSC: 14N10 Enumerative problems (combinatorial problems) in algebraic geometry 14H50 Plane and space curves 14L30 Group actions on varieties or schemes (quotients) Keywords:projective linear group; degree; stabilizer; blow-up; orbit closure; PGL(3)-orbit; enumerative geometry of plane curves PDFBibTeX XMLCite \textit{P. Aluffi} and \textit{C. Faber}, Ann. Inst. Fourier 50, No. 1, 151--196 (2000; Zbl 0953.14030) Full Text: DOI arXiv Numdam EuDML References: [1] [Alu] , The enumerative geometry of plane cubics I: smooth cubics, Trans. AMS, 317 (1990), 501-539. · Zbl 0703.14035 [2] [AF1] , , Linear orbits of smooth plane curves, J. Alg. Geom., 2 (1993), 155-184. · Zbl 0804.14015 [3] [AF2] , , Linear orbits of d-tuples of points in ℙ1, J. reine & angew. Math., 445 (1993), 205-220. · Zbl 0781.14036 [4] [AF3] , , A remark on the Chern class of a tensor product, Manu. Math., 88 (1995), 85-86. · Zbl 0863.14007 [5] [AF4] , , Plane curves with small linear orbits II, Preprint, math.AG/9906131. · Zbl 1100.14528 [6] [Ful] , Intersection Theory, Springer Verlag, 1984. · Zbl 0541.14005 [7] [Ghi] , 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 [8] [MX] , , Geometry of Complete Cuspidal Cubics, in Algebraic curves and projective geometry (Trento, 1988), Springer Lecture Notes in Math. 1389, 195-234. · Zbl 0688.14050 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.