When speaking about blow-ups, I've seen that everybody says "let $Y \subset X$ be a smooth closed subvariety of codimension $r$..."
What happens when one want to blow-up subvarieties which are not of pure dimension?
You can always decompose $Y$ into a union of pure dimension subvarieties and blow-up them successively. Does the order matter? Or the assumption of $Y$ being smooth mean that these components cannot intersect?