Let $$ X, Y \subset \mathbb{P}^N$$ be two non singular algebraic varieties of dimensions $k$ and $l$ that intersect transversally. Is it true that the ``dimension'' of the variety $\overline{X} \cap \overline{Y} - X\cap Y$ is strictly less than $k+l-N$, which is the dimension of $X\cap Y$ as a complex manifold. What I am worried about is that when you take the closure and then take intersections you may add singular things of very high dimension to $X\cap Y$.

I think it is true that the dimension of $\overline{X\cap Y}- X \cap Y$ is strictly less than $k+l-N$.

yourvarieties $X$ and $Y$ intersect transversally? Are you using Bertini's theorem? If so, then you can apply Bertini's theorem to the closures of $X$ and $Y$. -- Jason – Jason Starr Oct 23 '11 at 18:50