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Nef and Big Divisors

Algebraic Geometry
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Tsakanikas Nickos
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Nef and Big Divisors

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Post by Tsakanikas Nickos » Sat Nov 26, 2016 7:09 pm

Let $X$ be an (irreducible) projective variety and let $D$ be a divisor on $X$. Show that the following are equivalent:
  1. $D$ is nef and big.
  2. There exists an effective divisor $N$ and a $k_{0} \in \mathbb{N} $ such that $D - \frac{1}{k} N$ is an ample $ \mathbb{Q} $-divisor for all $k \geq k_{0}$.

Moreover, show that if we now assume $D$ to be a (nef and big) $\mathbb{R}$-divisor, then there exists an effective $ \mathbb{R} $-divisor $N$ and a $k_{0} \in \mathbb{N} $ such that $D - \frac{1}{k} N$ is an ample $ \mathbb{R} $-divisor for all $k \geq k_{0}$.
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