Abstract
We are interested in a particular geometry of plane curves in characteristicp>0, which was inspired by Thas's article [13]. We will prove that any plane curve of degree > 2 whose tangent lines at collinear points are concurrent is either a strange curve or projectively equivalent to the Fermat curve of degreeq + 1, whereq is a power ofp.
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References
Bayer, V. and Hefez, A.: Strange curves,Comm. Alg. 19 (1991), 3041–3059.
Deuring, M.: Invarianten und Normalformen elliptischer Funktionenkörper,Math. Z. 47 (1941), 47–56.
Hefez, A.: Non-reflexive curves,Compositio Math. 69 (1989), 3–35.
Hefez, A. and Kleiman, S. L.: Notes on duality of projective varieties, inGeometry Today, Roma 1984 (E. Arbarello, C. Procesi, E. Strickland, eds.),Prog. Math. 60, Birkhäuser, Boston, 1985, pp. 143–184.
Hefez, A. and Vainsencher, I.: Strange plane curves,Comm. Alg. 19 (1991), 333–345.
Homma, M.: Funny plane curves in characteristicp>0,Comm. Alg. 15 (1987), 1469–1501.
Homma, M.: A souped-up version of Pardini's theorem and its application to funny curves,Compositio Math. 71 (1989), 295–302.
Kaji, H.: On the Gauss maps of space curves in characteristicp>0,Compositio Math. 70 (1989), 177–197.
Kleiman, S. L.: The enumerative theory of singularities, inReal and Complex Singularities, P. Holm (ed.),Proc. Conf. Oslo 1976, Stijhoff & Noordhoof, Groningen, 1977, pp. 297–396.
Kleiman, S. L.: Tangency and duality, inProc. 1984 Vancouver Conf. in Algebraic Geometry, J. Carrell, A. V. Geramita and P. Russell (eds.),CMS Conf. Proc. 6, Amer. Math. Soc., Providence, 1986, pp. 163–226.
Kleiman, S. L.: Multiple tangents of smooth plane curves (after Kaji) inProc. 1988 Sundance Conf. on Algebraic Geometry, B. Harbourne and R. Speiser (eds.),Contemporary Math. 116, Amer. Math. Soc., Providence, 1991, pp. 71–84.
Stöhr, K. O. and Voloch, J. F.: Weierstrass points and curves over finite fields,Proc. London Math. Soc. 52 (1986), 1–19.
Thas, J. A.: A combinatorial characterization of Hermitian curves,J. Algebra Combin. 1 (1992), 97–102.
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Homma, M. Plane curves whose tangent lines at collinear points are concurrent. Geom Dedicata 53, 287–296 (1994). https://doi.org/10.1007/BF01264002
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DOI: https://doi.org/10.1007/BF01264002