Let $X$ be a projective surface over $\mathbb{C}$, let $x\in X$ be the only singular point of $X$. Let $L$ be an ample line bundle on $X$. Consider the blow up $Y$ of $X$ along $x$, $f:Y\longrightarrow X$. Let $L'$ be the pull back of $L$ to $Y$ and let $E$ be the exceptional divisor.

We have the following short exact sequence on $Y$,

$0\longrightarrow L'-E\longrightarrow L'\longrightarrow L'|_E\longrightarrow 0$.

So there is an injection $|L'-E|\hookrightarrow |L'|$ which sends $C\mapsto C+E$.

1) Now how do we describe curves in $|L'-E|$? Are these curves that contain $E$ or curves that pass through a point in $E$? Do they have to intersect $E$?

2) What does the divisor $C+E$ mean? The divisor $C+E$ comes from $X$ which passes through $x$?

3) Is $C$ is a closed subscheme of $C+E$. Then we have the surjection $\mathcal{O}_{C+E} \longrightarrow \mathcal{O}_{C}$. What is the ideal sheaf? It looks to me to be $\mathcal{O}_E$.

4) If $C$ is indeed a closed subscheme of $C+E$, and we start with a line bundle $A$ on $C+E$, and call the pullback to $C$ as $A'$, the degrees will be same I suppose. But what is the relationship between $h^0(C,A')$ and $h^0(C+E,A)$?

Sorry about the long post. These questions have been bothering me for a while. Thanks in advance!

linearly equivalentto one, say $D$, with non-negative coefficients. In that case $D\in |O(C-E)|$ corresponds, as above to a curve passing through $x$. 3),4) If $C$ is a curve (irreducible!) then $C-E$ is not a subscheme of $C$. The only way one can make sense of these statements is in the case when $C$ itself is a reducible divisor, and $E$ is one of the components. Then some $C-mE$ could mean the divisor corresponding to the other components of $C$. $\endgroup$ – Walter Neff Oct 30 '15 at 17:23