I'd like to blow-up the two diagonals $X=V(X_1-X_2)\cup V(X_1+X_2)$ (in $\mathbb{A}^2_k$) with center $\mathcal{O}=(0,0)$. I had the idea to use the blowup of the plane so that I obtain $\widetilde{X}\subseteq\mathbb{A}^2_k\times\mathbb{P}^1_k$ with equations
$$ \left\lbrace\begin{array}{l} X_1^2-X_2^2&=0 \\\ X_1T_2-X_2T_1&=0 \end{array}\right. $$
But I had the idea that the blowup of the diagonals should be two disjoints lines and here it seems not to be the case because the exceptionnal divisor is $\mathbb{P}^1_k$ but it should be two points.
I guess my error is to apply the blowup of the plane to my subvariety $X$ But how to do without that? I have the construction with $$ \text{Proj}(\oplus I^n ) $$ with $I=(X_1,X_2)$, but it give me the same thing because with $S=k[X_1,X_2]/(X_1^2-X_2^2)$ I get $$\begin{align*} \text{Proj}(\oplus I^n ) &= \text{Proj}(S[T_1,T_2]/(T_1X_2-X_1T_2)) \\\ &= \text{Proj}(S\otimes_k k[T_1,T_2])/(T_1X_2-T_2X_1) \end{align*}$$ so that $\widetilde{X}$ appears as the closed subscheme of $X\times_k \mathbb{P}^1_k$ given by equation $T_1X_2-T_2X_1=0$ and for $X_1=X_2=0$ I always has the same fiber and my two lines are not disjoints?