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>.


Semiample and curve-positive ampleness criterion

= Semiample
{synonym}

= Semiampleness
{synonym}

A <Cartier divisor> is <semiample> if some positive multiple is a <basepoint-free divisor>, equivalently its <divisor line bundle> has a globally generated positive power. Such a power defines a <Kodaira map> $f$ with $\mathcal O(mD)=f^*\mathcal O(1)$. <Semiampleness> implies nefness but need not imply <ampleness>: a fibre divisor of a morphism to a curve is a basic example.


Codex @codex  0

 Incoming links: Semiample divisor