Let $X$ be a smooth projective surface over $\mathbb{C}$ with a fixed embedding in some $\mathbb{P}^n$, and let $\mathcal{O}_X(1)$ be the corresponding very ample line bundle. My question is:
Is it true that for $d> 0$, $\mathcal{O}_X(d)$ is the line bundle corresponding to a divisor $D$ obtained by intersecting $X$ with a degree $d$ hypersurface in $\mathbb{P}^n$? I ask this to make sense of the linear system |$\mathcal{O}_X(d)$|, as I have only seen linear system for divisors of the form $\mathcal{O}_X(D)$ for some divisor $D$.
Is it also true that the curves in $X$ in the linear system |$\mathcal{O}_X(d)$| are also obtained in the same way, i.e. by intersecting with degree $d$ hypersurfaces?
Thanks in advance.
I am not sure where to start for 1. So, let me try. If $Y\subset X$ is a closed subset, both irreducible and smooth (though smoothness is not essential) and if $L$ is a line bundle on $X$ we can restrict $L$ to $Y$ to get a line bundle $L_{|Y}$. If $L=\mathcal{O}_X(D)$ for a divisor $D\subset X$ and $Y$ is not contained in $D$, then $L_{|Y}=\mathcal{O}_Y(D\cap Y)$, where of course the intersection is scheme-theoretic.
For 2, you have a map $H^0(X,L)\to H^0(Y,L_{|Y})$. Non-zero elements of the first correspond to effective divisors $D$ with $\mathcal{O}_X(D)=L$ and similarly for the second. So, if this map is not surjective, there would be effective divisors $E$ on $Y$ with $\mathcal{O}_Y(E)=L_{|Y}$, but not coming as intersection with $Y$ of a divisor from $X$. A typical example of such for surfaces would be an embedding of $\mathbb{P}^2\subset\mathbb{P}^5$, using $\mathcal{O}_{\mathbb{P}^2}(2)$. If you have not seen this, please look it up in Veronese embeddings, secant varieties etc.