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Stability of Frobenius pull-backs of tangent bundles and generic injectivity of Gauss maps in positive characteristic

Authors:	ATSUSHI NOMA

Institution:	(1) Department of Mathematics, Faculty of Education, Yokohama National University, 79-2 Tokiwadai, Hodogaya-ku, Yokohama, 240, Japan

Abstract:	For a smooth projective variety X of dimension n in a projective space $\mathbb{P}^N$ defined over an algebraically closed field k, the Gauss mapis a morphism from X to the Grassmannian of n-plans in $\mathbb{P}^N$ sending $x \in X$ to the embedded tangent space $T_x X \subset \mathbb{P}^N$ .The purpose of this paper is to prove the generic injectivity of Gauss mapsin positive characteristic for two cases; (1) weighted complete intersectionsof dimension $n \geqslant 3$ of general type; (2) surfaces or 3-folds with -semistable tangent bundles; based on a criterion of Kaji by looking atthe stability of Frobenius pull-backs of their tangent bundles. The first result implies that a conjecture of Kleiman--Piene is true in case X is of general type of dimension $n \geqslant 3$ . The second result is a generalization of the injectivity for curves.

Keywords:	Gauss map tangent bundle stable vector bundle
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