I'm trying to understand the following result:
Let $S$ be a smooth, projective surface over $\mathbb{C}$ and let $D$ be a divisor on $S$. Let $H$ be a hyperplane section of $S$ for a given embedding. Then for some $n \geq 0$, we can write $D \equiv A - B$, where $A$ and $B$ are smooth curves on $S$ with $A \equiv D + nH$ and $B \equiv nH$.
OK, I can sort of see why "twisting" enough will make $D + nH$ "effective", even though I'm not sure of the details - the invertible sheaf corresponding to $H$ is very ample, so I guess the sheaf corresponding to $D + nH$ will be very ample too for $n$ large enough (right?). This means that $D + nH$ will be linearly equivalent to a curve $A$ (not necessarily irreducible) on $S$, and so will $B$.
But where does the smoothness come from?