The Michigan Mathematical Journal

On Separable Higher Gauss Maps

Katsuhisa Furukawa and Atsushi Ito

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We study the mth Gauss map in the sense of F. L. Zak of a projective variety XPN over an algebraically closed field in any characteristic. For all integers m with n:=dim(X)m<N, we show that the contact locus on X of a general tangent m-plane is a linear variety if the mth Gauss map is separable. We also show that for smooth X with n<N2, the (n+1)th Gauss map is birational if it is separable, unless X is the Segre embedding P1×PnP2n1. This is related to Ein’s classification of varieties with small dual varieties in characteristic zero.

Article information

Michigan Math. J., Volume 68, Issue 3 (2019), 483-503.

Received: 24 June 2017
Revised: 30 October 2017
First available in Project Euclid: 18 April 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14N05: Projective techniques [See also 51N35]


Furukawa, Katsuhisa; Ito, Atsushi. On Separable Higher Gauss Maps. Michigan Math. J. 68 (2019), no. 3, 483--503. doi:10.1307/mmj/1555574416.

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  • [1] E. Ballico, On the Gauss maps of singular projective varieties, J. Aust. Math. Soc. 72 (2002), 119–130.
  • [2] C. Ciliberto and K. Hulek, A bound on the irregularity of Abelian scrolls in projective space, Complex geometry (Göttingen, 2000), 85–92, Springer, Berlin, 2002.
  • [3] L. Ein, Varieties with small dual varieties, I, Invent. Math. 86 (1986), 63–74.
  • [4] G. Fischer and J. Piontkowski, Ruled varieties, Advanced Lectures in Mathematics, Friedr. Vieweg & Sohn, Braunschweig, 2001.
  • [5] S. Fukasawa, Developable varieties in positive characteristic, Hiroshima Math. J. 35 (2005), 167–182.
  • [6] S. Fukasawa, Varieties with non-linear Gauss fibers, Math. Ann. 334 (2006), 235–239.
  • [7] S. Fukasawa, On Kleiman–Piene’s question for Gauss maps, Compos. Math. 142 (2006), 1305–1307.
  • [8] S. Fukasawa, A remark on Kleiman–Piene’s question for Gauss maps, Comm. Algebra 35 (2007), 1201–1204.
  • [9] K. Furukawa, Duality with expanding maps and shrinking maps, and its applications to Gauss maps, Math. Ann. 358 (2014), 403–432.
  • [10] K. Furukawa and A. Ito, Gauss maps of toric varieties, Tohoku Math. J. (2) 69 (2017), 431–454.
  • [11] K. Furukawa and A. Ito, On Gauss maps in positive characteristic in view of images, fibers, and field extensions, Int. Math. Res. Not. 2017, no. 8, 2337–2366.
  • [12] P. Griffiths and J. Harris, Algebraic geometry and local differential geometry, Ann. Sci. École Norm. Sup. (4) 12 (1979), 355–432.
  • [13] A. Hefez and S. Kleiman, Notes on the duality of projective varieties, Geometry today (Rome, 1984), Progr. Math., 60, 143–183, Birkhäuser Boston, Boston, MA, 1985.
  • [14] T. A. Ivey and J. M. Landsberg, Cartan for beginners: differential geometry via moving frames and exterior differential systems, Graduate Studies in Mathematics, 61, American Mathematical Society, Providence, RI, 2003.
  • [15] H. Kaji, On the tangentially degenerate curves, J. Lond. Math. Soc. (2) 33 (1986), 430–440.
  • [16] H. Kaji, On the Gauss maps of space curves in characteristic $p$, Compositio Math. 70 (1989), 177–197.
  • [17] H. Kaji, On the duals of Segre varieties, Geom. Dedicata 99 (2003), 221–229.
  • [18] H. Kaji, Higher Gauss maps of Veronese varieties–a generalization of Boole’s formula and degree bounds for higher Gauss map images, Comm. Algebra 46 (2018), 4064–4078.
  • [19] S. L. Kleiman, Plane forms and multiple-point formulas, Algebraic threefolds (Varenna, 1981), Lecture Notes in Math., 947, 287–310, Springer, Berlin-New York, 1982.
  • [20] S. L. Kleiman, Tangency and duality, Proceedings of the 1984 Vancouver conference in algebraic geometry, CMS Conf. Proc., 6, 163–226, Amer. Math. Soc., Providence, RI, 1986.
  • [21] S. L. Kleiman and R. Piene, On the inseparability of the Gauss map, Enumerative algebraic geometry (Copenhagen, 1989), Contemp. Math., 123, 107–129, Amer. Math. Soc., Providence, RI, 1991.
  • [22] R. Lazarsfeld, Positivity in algebraic geometry I, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 48, Springer-Verlag, Berlin, 2004.
  • [23] A. Noma, Gauss maps with nontrivial separable degree in positive characteristic, J. Pure Appl. Algebra 156 (2001), 81–93.
  • [24] J. Rathmann, The uniform position principle for curves in characteristic $p$, Math. Ann. 276 (1987), 565–579.
  • [25] J. C. Sierra, A remark on Zak’s theorem on tangencies, Math. Res. Lett. 18 (2011), 783–789.
  • [26] R. Speiser, Vanishing criteria and the Picard group for projective varieties of low codimension, Compositio Math. 42 (1980/81), 13–21.
  • [27] A. H. Wallace, Tangency and duality over arbitrary fields, Proc. London Math. Soc. (3) 6 (1956), 321–342.
  • [28] F. L. Zak, Tangents and secants of algebraic varieties, Translations of Mathematical Monographs, 127, American Mathematical Society, Providence, RI, 1993.