6
$\begingroup$

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?

$\endgroup$
5
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.