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On the inseparable degree of the Gauss map of higher order for space curves. (English) Zbl 0752.14032
Let $$X$$ be a non-degenerate curve in the projective space $$\mathbb{P}^ N$$ over an algebraically closed field of characteristic $$p>0$$. The authors determine the inseparable degree of the Gauss map defined by the osculating $$m$$-planes of $$X$$. More precisely, let $$x$$ be a general point of $$X$$, $$H$$ a general hyperplane containing the osculating $$m$$-plane to $$X$$ at $$x$$, and $$b_{m+1}$$ the intersection multiplicity of $$H$$ and $$X$$ in $$x$$. Then the inseparable degree of the Gauss map of order $$m$$ equals the highest power of $$p$$ dividing $$b_{m+1}$$.

##### MSC:
 14H50 Plane and space curves
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##### References:
 [1] A. Garcia and J. F. Voloch: Duality for projective curves (preprint). · Zbl 0766.14021 · doi:10.1007/BF01237362 [2] A. Hefez: Letter to M. Homma dated 19 September 1990. · Zbl 0974.51001 [3] A. Hefez and J. F. Voloch: Frobenius non classical curves. Arch. Math., Basel, 54, 263-273 (1990). · Zbl 0662.14016 · doi:10.1007/BF01188523 [4] M. Homma: Funny plane curves in characteristic p>0. Comm. Algebra, 15, 1469-1501 (1987). · Zbl 0623.14014 · doi:10.1080/00927878708823481 [5] H. Kaji: On the Gauss maps of space curves in characteristic p. Compositio Math., 70, 177-197 (1989). · Zbl 0692.14015 · numdam:CM_1989__70_2_177_0 · eudml:89960 [6] H. Kaji: On the inseparable degrees of the Gauss maps and the projection of the conormal variety to the dual of higher order for space curves. Math. Ann. (to appear). · Zbl 0734.14017 · doi:10.1007/BF01444633 · eudml:164926 [7] D. Laksov: Wronskians and Plucker formulas for linear systems on curves. Ann. Sci. Ecole Norm. Sup., (4) 17, 45-66 (1984). · Zbl 0555.14008 · numdam:ASENS_1984_4_17_1_45_0 · eudml:82136 [8] F.K.Schmidt: Die Wronskische Determinante in Beliebigen differenzierbaren Funktionenkorpern. Math. Z., 45, 62-74 (1939). · Zbl 0020.10201 · doi:10.1007/BF01580273 · eudml:168839 [9] K. 0. Stohr and J. F. Voloch: Weierstrass points and curves over finite fields. Proc. London Math. Soc, (3) 52, 1-19 (1986). · Zbl 0593.14020 · doi:10.1112/plms/s3-52.1.1
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