The separability of the Gauss map and the reflexivity for a projective surface |
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Authors: | Satoru Fukasawa Hajime Kaji |
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Institution: | (1) Department of Mathematics, Graduate School of Science, Hiroshima University, Kagamiyama 1-3-1, Higashi-Hiroshima, Hiroshima 739-8526, Japan;(2) Department of Mathematical Sciences, School of Science and Engineering, Waseda University, Ohkubo 3-4-1, Shinjuku, Tokyo 169–8555, Japan |
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Abstract: | It is known that if a projective variety X in P
N
is reflexive with respect to the projective dual, then the Gauss map of X defined by embedded tangent spaces is separable, and moreover that the converse is not true in general. We prove that the
converse holds under the assumption that X is of dimension two. Explaining the subtleness of the problem, we present an example of smooth projective surfaces in arbitrary
positive characteristic, which gives a negative answer to a question raised by S. Kleiman and R. Piene on the inseparability
of the Gauss map.
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Keywords: | Gauss map Reflexive Separable |
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