Welcome to mathimatikoi.org forum; Enjoy your visit here.

Nef and Big Divisors

Algebraic Geometry
Tsakanikas Nickos
Community Team
Articles: 0
Posts: 314
Joined: Tue Nov 10, 2015 8:25 pm

Nef and Big Divisors

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 $k \geq k_{0}$.