When reading a book, I encountered the following assertion: let $k$ be a field (maybe perfect if it makes things easier), $X$ a smooth $k$-variety and $Y$ a closed irreducible subvariety, also smooth. Let $\widetilde{X}$ be the blow-up of $X$ along $Y$ whose exceptional divisor is denoted by $E$. Let $D$ be an effective Cartier divisor with normal crossings whose irreducible components are $(D_i)_{i \in I}$. Denote by $(\widetilde{D}_i)$ their strict transforms w.r.t the blow-up along $Y$. Suppose that $Y \not\subset D_i \forall i\in I$ and $\sum_{i \in I} D_i$ intersects $Y$ transversally, then why
- $\widetilde{D}_i$ are irreducible.
- $E + \sum_{i \in I}\widetilde{D}_i$ is a divisor with normal crossings in $\widetilde{X}$ . There is a similar question here on MO but the answer is neither clear nor complete for me. I expect some local computations.
For 1) I think it could be paraphrased as: given $Y \subset X$ with $X$ irreducible, then $\mathrm{Bl}_Y(X)$ is irreducible, this would be the case if $X \setminus Y \subset \mathrm{Bl}_Y(X)$ is dense in $\mathrm{Bl}_Y(X)$ but I don't know how to proceed.
For 2), I may assume that $X = \mathrm{Spec}(A)$ with $A$ regular local and $Y$ is cut out by $I \leq A$ an ideal generated by a part $(f_1,...,f_r)$ of a regular system of parameters $(f_1,...,f_d)$ ($r \leq d$), then $\widetilde{X}$ is $$\mathrm{Bl}_Y(X) = \widetilde{X} = \mathrm{Proj}(A[T_1,...,T_r]/(f_iT_j - f_jT_i)_{1 \leq i,j \leq r})$$ and the exceptional divisor is obtained by dividing out $I$ and taking $\mathrm{Proj}$, which is $$E = \mathrm{Proj}((A/I)[T_1,...,T_r]),$$ so the problem is, from the beginning we can assume that each $D_i$, is cut out by some $f_i$ right? (the condition $Y \not\subset D_i$ is there to make sure that the equations $D_i = \left \{f_i = 0 \right \}$ is not any of $(f_1,...,f_r)$ I guess).
I hope that someone could provide a complete computation and a clear argument. Thanks in advance.