= Semiample and curve-positive ampleness criterion
A <semiample divisor> on an integral <projective variety> is <ample> if it has positive degree on every integral <projective curve>. Its <Kodaira map> cannot contract a positive-dimensional fibre, since such a fibre contains a curve and the pulled-back hyperplane bundle has degree zero there. Thus the morphism is proper and a <quasi-finite morphism>, hence a <finite morphism>. The <finite pullback of an ample line bundle> is <ample>.
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