Let $X$ be an algebraic variety (not necessarily projective) over $\mathbb{C}$, and $V\_1,V\_2\subset X$ two projective subvarieties of $X$, with $\textrm{codim}(V\_1)=\textrm{codim}(V\_2)=2$. Suppose $V\_1$ and $V\_2$ are birational equivalent. Denote by $X\_1$ and $X\_2$ the blow up of $X$ along $V\_1$ and $V\_2$ respectively, then what can be said about the geometry and topology of $X\_1$ and $X\_2$? For example, suppose $\dim(X)=n$, is it necessarily true that $H^n(X\_1,\mathbb{Z})\cong H^n(X\_2,\mathbb{Z})$?



Blowing up along birational equivalent subvarieties
Let us give an answer in the case where $X$ is rational. Since $V\_1$ and $V\_2$ are birationally equivalent and of codimension $2$, there exists a birational map of $X$ which restricts to a birational map from $V\_1$ onto $V\_2$. (See "Equivalent birational embeddings" of Mella and Polastri). Hence, if $Y\_1\to X$ and $Y\_2\to X$ are the blow-ups of $V\_1$ and $V\_2$, there is a birational map $Y\_1\dashrightarrow Y\_2$ which sends the exceptional divisor $E\_1$ onto the exceptional divisor $E\_2$. This means that the pairs $(Y\_1,E\_1)$, $(Y\_2,E\_2)$ are birational. In particular, they have the same Kodaira dimension, but also share many birational geometric properties.


