Abstract
We investigate singularities of a general plane section of the dual variety of a smooth projective variety, or more generally, the discriminant variety associated with a linear system of divisors on a smooth projective variety. We show that, in characteristic 3, singular points of E 6-type take the place of ordinary cusps in characteristic 0.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Artin, M. Coverings of the rational double points in characteristic p, Complex Analysis and Algebraic Geometry, pp. 11–22. Iwanami Shoten, Tokyo, (1977).
Dolgachev, I, Libgober, A. On the Fundamental Group of the Complement To a Discriminant Variety, Algebraic geometry (Chicago, Ill., 1980), Lecture Notes in Math., vol. 862, pp. 1–25. Springer, Berlin (1981).
Fulton, W. Intersection Theory, 2nd ed, Ergebn. Math. Grenzgeb. 3. Folge. A Series of Modern Surveys in Mathematics , vol. 2. Springer-Verlag, Berlin (1998).
Greuel G.-M., Kröning H. (1990). Simple singularities in positive characteristic. Math. Z. 203(2): 339–354
Harris J., Tu L.W. (1984). On symmetric and skew-symmetric determinantal varieties. Topology 23(1): 71–84
Hefez A., Thorup A. (1987). Reflexivity of Grassmannians and Segre varieties. Comm. Algebra 15(6): 1095–1108
Hefez A. (1989). Nonreflexive curves. Compositio Math. 69(1): 3–35
Hefez, A., Kleiman, S.L. Notes on the Duality Of Projective Varieties. Geometry today (Rome, 1984), Progr. Math., vol. 60, pp. 143–183. Birkhäuser, Boston, MA (1985).
Homma M. (1993). On duals of smooth plane curves. Proc. Amer. Math. Soc. 118(3): 785–790
Józefiak T., Lascoux A., Pragacz P. (1981). Classes of determinantal varieties associated with symmetric and skew-symmetric matrices. Izv. Akad. Nauk SSSR Ser. Mat. 45(3): 662–673
Kaji H. (1989). On the Gauss maps of space curves in characteristic p. Compositio Math. 70(2): 177–197
Kaji H. (2003). On the duals of Segre varieties, Geom. Dedicata 99, 221–229
Kaji H., Noma A. (1997). On the generic injectivity of the Gauss map in positive characteristic. J. Reine Angew. Math. 482, 215–224
Katz, N.M. Pinceaux de Lefschetz: théorème d’existence. Groupes de monodromie en géométrie algébrique. II, Springer-Verlag, Berlin (1973). Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 II), Dirigé par P. Deligne et N. Katz, Lecture Notes in Mathematics, vol. 340, pp. 212–253.
Kleiman, S.L. The enumerative theory of singularities. Uspekhi Mat. Nauk 35(6), 69–148 (1980) Translated from the English by V.S. Kulikov.
Kleiman S.L. (1984). About the conormal scheme. Complete intersections (Acireale, 1983), Lecture Notes in Math., vol. 1092, pp. 161–197. Springer, Berlin
Kleiman, S.L. Tangency and duality. In: Proceedings of the 1984 Vancouver conference in algebraic geometry (Providence, RI), CMS Conf. Proc., vol. 6, pp. 163–225. Amer. Math. Soc. (1986).
Libgober A.S., Wood J.W. (1990). Uniqueness of the complex structure on Kähler manifolds of certain homotopy types. J. Differential Geom. 32(1): 139–154
Pardini R. (1986). Some remarks on plane curves over fields of finite characteristic. Compositio Math. 60(1): 3–17
Pragacz, P. Cycles of Isotropic Subspaces And Formulas For Symmetric Degeneracy Loci. Topics in algebra, Part 2 (Warsaw, 1988), Banach Center Publ., vol. 26, pp. 189–199. PWN, Warsaw (1990).
Seshadri, C.S. L’opration de Cartier. Applications. Séminaire C. Chevalley, 3ième année: 1958/59, pp. 83–135. Variétés de Picard, École Normale Supérieure, Paris (1960).
Shimada, I. Singularities of dual varieties in characteristic 2, preprint. http://www.math.sci.hokudai.ac.jp/~shimada/preprints.html
Shimada, I. Lattices of algebraic cycles on Fermat varieties in positive characteristics. Proc. London Math. Soc. (3) 82(1), 131–172 (2001).
Wallace, A.H. Tangency and duality over arbitrary fields. Proc. London Math. Soc. (3) 6, 321–342 (1956).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Shimada, I. Singularities of Dual Varieties in Characteristic 3. Geom Dedicata 120, 141–177 (2006). https://doi.org/10.1007/s10711-006-9074-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10711-006-9074-z