Let $S\subset\mathbb{P}^2$ be a finite set of points in the plane, consider the blow-up of $\mathbb{P}^2$ along $S$: $$X=Bl_S(\mathbb{P^2})\rightarrow\mathbb{P}^2$$ and denote the divisor $D=2H-E$, where $E$ is the exceptional divisor and $H$ is the pull back of a line in $\mathbb{P}^2$.
This is in Lazarsfeld's Positivity in Algebraic Geometry I, example 2.2.13. He said that by taking $S$ to be a large number of sufficiently general points, one can assume that there are no curves of degree $2m$ in $\mathbb{P}^2$ having multiplicity $\geq m$ at each of the points of $S$.
So my first question is why there are no such curves? I apologize that this would be a stupid question since I think the degree of a curve may bounds its singularity, but I can't give a proof.
One the other hand, Lazarsfeld said that if the points of $S$ are colinear, then $D-H$ is effective, and hense $D$ is big.
My second question is how can we show the bigness of $D$, although $D-H$ is effective, $H$ is not ample, isn't it?
I would greatly appreciate any comments or answers, thank you!