TITLE: Does a closed immersion of an affine scheme in a smooth scheme factor over an open affine subscheme?
QUESTION [5 upvotes]: Assume we have an affine scheme $A$ that comes with a closed immersion into a smooth scheme $i \colon A \hookrightarrow M$, which is not necessarily affine. 
Does there exist an affine open subscheme $j \colon V \hookrightarrow M$ such that $A$ already embeds in $V$?
Expressed in Diagrams, does there exist an affine open subscheme $V$ of $M$ such that we have a factorization

     i
A (----> M
 \      /
  \    /
     V ?

REPLY [5 votes]: There are also examples of smooth proper (nonprojective) varieties $V$ with a finite subset not contained in any affine open subset.