Take the tour ×  Let $Y$ be a normal surface and let $X$ be a
closed subscheme of codimension 2, i.e., $X$ is a finite set of
closed points.  Let $D$ be a Weil divisor on $Y$.  Question.
Does there exist a Weil divisor $E$ on $Y$ which is linearly
equivalent to $D$ and does not go through $X$? (Edit: I do not
assume $E$ to be effective.)  Of course, when $D$ does not go
through $X$ the answer is yes.  When I say that $E$ does not go
through $X$, I mean that the support of $E$ and $X$ are
disjoint.  I'm interested in this question in the most general
set-up known. For example, $Y$ is an integral noetherian
separated excellent normal 2-dimensional scheme. If you prefer
sticking to algebraic surfaces, I would be glad to hear about
what's possible in that case too.  The motivation for this
question comes from an article by Mumford in which he defines an
intersection pairing on a normal surface. In this case $X$ is
the singular locus of $Y$.  share|improve this question add
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