Nef and Big Divisors
Posted: 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:
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 $k \geq k_{0}$.
- $D$ is nef and big.
- 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 $k \geq k_{0}$.