Shokurov's non-vanishing theorem in 1-dimension

Question

The Shokurov's non-vanishing theorem states the following:

Let $(X,B)$ be a projective klt pair and $L$ a nef Cartier divisor on $X$. Suppose there is some $a>0$ such that $aL-(K_X+B)$ is ample. Then $H^0(X,nL)\neq 0$ for all $n\gg 0$.

Many of textbooks (for example, Kollár-Mori) prove the theorem by using induction on dimension of $X$. To my knowledge, in order to use induction, one must first prove the case when $\dim X=1$, but it seems that the books lack of explanation on how to prove the theorem when $\dim X=1$, and I also don't know how.

I tried to solve the question by using the Riemann-Roch theorem, but then I encountered another question: does $L$ is nef implies $\deg(nL)\ge 0$ for all $n\gg 0$?

Any clarification would helps.

Answer 1

Nef means non-negative degree on every integral curve, so a line bundle $L$ on an integal curve $X$ is nef iff $\deg L \geq 0$. Therefore $L$ nef implies $\deg nL \geq 0 $ for all $n>0$.

To reference the definition of a nef line bundle, see Wikipedia, Vakil section 19.4.8, or similar. (This should be in many texts, these are just the two that I have available to check on at this moment.)