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!



Relation between curves in a complete linear system contained in another
In the following I suppose that by "curve" in a linear system you mean "effective divisor" vithout any claim about being irreducible and/or reduced.


1) Curves in |L'| are exactly the pull-back of curves in |L|. So "curves" in |L'-E| are obtained by pulling-back curves in L through x, and then remove $E$ with multiplicity 1. That's usually the strict transform, but may contain $E$ if the curve in $|L|$ passes through $x$ with high multiplicity.


2) $C+E$ is the reducible effective divisor union of $C$ and $E$ (of course) and equals the pull-back on $Y$ of the image of $C$ o $X$. In other words you get all pull-backs of curves in $|L|$ passing through $x$.


3) No, it is ${\mathcal O}\_E(-C)$, its sections are supported on $E$ but vanish on the intersection with $C$.


4) No, the degree do not need to be the same. Assume for instance that $C$ is smooth irreducible intersecting transversally $E$ in $d$ points. Then every pair of line bundles on $C$ and $E$ may be "glued" to a line bundle on $C+E$.


