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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?

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1 Answer 1

up vote 5 down vote accepted

As you note, (the sheaf corresponding to) $D + nH$ will be very ample for $n$ large enough. Bertini then implies that a generic member of the linear system $|D+nH|$ will be smooth. Similary, since $nH$ is very ample, a generic member of $|nH|$ will be smooth. Hence $D$ is linearly equivalent to a difference of smooth curves.

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