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 $H^n(X_1,\mathbb{Z})\cong H^n(X_2,\mathbb{Z})$?

4$\begingroup$ $H^n(X_i,Z) = H^n(X,Z) \oplus H^{n2}(V_i,Z)$, so if you want them to be equal for the blowups, you should have an equality for the centers which is not necessary true if $V_1$ and $V_2$ are only birational. $\endgroup$– SashaSep 9, 2014 at 7:06
1 Answer
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 blowups 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.