I have one more question about the Example (I.5.1) on page 7 from Rick Miranda's the basic theory of elliptic surfaces:

Let $C_1$ be a smooth cubic curve in $\mathbb{P^2}$ and let $C_2$ be any other cubic. Let $F_1, F_2 \in \mathbb{C}[X,Y,Z]$ the homogeneous cubic polynomials generating the vanishing ideals of $C_1$, respectively $C_2$. With intersection theory and Bezout's lemma these curves intersect in $9$ points. We form a pencil $X \subset \mathbb{P^2} $ generated by $C_1$ and $C_2$; in detail the pencil is defined as the union

$$ X = \bigcup_{[\lambda: \mu] \in \mathbb{P}^1} V(\lambda F_1 + \mu F_2) $$

of subschemes $V(\lambda F_1 + \mu F_2) $ in $\mathbb{P^2}$ running over $\mathbb{P}^1$. This gives only a rational map to $\mathbb{P^1}$, since this map is not defined in the nine intersection points of $C_1$ and $C_2$. Next is claimed that after blowing up $X$ in these points, we obtain a honest morphism $\pi: \widetilde{X} \to \mathbb{P^1}$ where $\widetilde{X}= \text{Bl}(X)_{x_1,..., x_9}$ is the blowup of the pencil $P$ in these $ 9 $ points.

Question: how can I calculate "by hand" the blow-up scheme $\widetilde{X}$ of $X$ along these nine points?

I know how to construct basically a blowup of $X$ along $Z \subset X$ corresponding to quasi-coherent sheaf of ideals
$ \mathcal{J} \subset \mathcal{O}_ X $ *assuming !!!* I know the structure sheaf $\mathcal{O}_ X $ of $X$ and sheaf of ideals
$ \mathcal{J}$.

In this case the most general way to construct the blowup is by

$$ \text{Bl}_Z(X):= \text{Proj}_X (\bigoplus_{n \ge 0} \mathcal{J}^n) $$

endowed with canonical projection $p: \text{Bl}_Z(X) \to X$ and with exceptional divisor $p^{-1}(Z)$. Since this construction behaves well with respect taking affine covers $(U_i =\text{Spec} A_i)_{i \in I}$ it is also possible construct it for these affine pieces $ U_i= \text{Spec} A_i $, ideal $J $ of $A_i$ and closed $Z = \text{Spec}(A_i/J) = U_i \cap Z \subset U_i$ separately by setting

$$ \text{Bl}_Z(U_i):= \text{Proj}(\bigoplus_{n \ge 0} I^n) $$

and gluing after that these pieces together.

Moreover, if we consider $X
\subset \mathbb{P}^n$ as sitting as closed subscheme inside
a projective space and want to blowup $X$ along
a subscheme $Y \subset X$ *assuming* that I know the associated
sheaf ideals $\mathcal{I} \subset
\mathcal{J} $ of $X$ respectively $Y$, then it can be done even more concrete:

As before since blowups behave well with respect to affine covers we can work locally by passing to any affine chart $\mathbb{A}^n$ of $\mathbb{P}^n$ and glue at the end the pieces together. Therefore we land in affine the situation where $ \operatorname{Spec}(A/J)=Y \cap \mathbb{A}^n \subset \operatorname{Spec}(A)=X \cap \mathbb{A}^n \subset \mathbb{A}^n$ where $A= \mathbb{C}[x_1,.., x_n]/I$ and and there exist an finitely generated ideal $\widetilde{J} = (g_1,..., g_m) \subset \mathbb{C}[x_1,.., x_n]$ such that $I \subset \widetilde{J} $ with $J = \widetilde{J}/I$.

We ignore $X$ for the moment define for each $g_i$ the ring $\mathbb{C}[x_1,.., x_n, g_1/g_i,..., g_m/g_i]$ which becomes later the $i$-th affine chart of the blowup. Since this ring contains $\mathbb{C}[x_1,.., x_n]$ the ideal $\widetilde{I}:= I \cdot \mathbb{C}[x_1,.., x_n, g_1/g_i,..., g_m/g_i]$ makes sense and we can form the quotient $B_i:=\mathbb{C}[x_1,.., x_n, g_1/g_i,..., g_m/g_i]/\widetilde{I}$. We glue the $B_i$'s together and we obtain the blowup of the affine piece $X \cap \mathbb{A}^n =\operatorname{Spec}(A) $ along $Y$. Then be glue second time the obtained blowups obtain over the affine charts of $ \mathbb{P}^n$ toghether.

To carry out that all 'by hand' is of course very laborious, but at the end of the day we have constucted explicitly a blowup of $X$ along $Y$.

*Back* to the blowup of the pencil
$X= \bigcup_{[\lambda: \mu] \in \mathbb{P}^1} V(\lambda F_1 + \mu F_2)
\subset \mathbb{P}^2$ along nine points $Y:= \{p_1,..., p_9 \}$
in Miranda's notes. I not know how to calculate here the blowup
of $X$ along $Y$ explicitly, since the pencil $X \subset \mathbb{P}^2$ is not described explicitly by an
associated ideal sheaf determining the structure sheaf $\mathcal{O}_X$ which
is neccessary to know for the calculation of the blowup by all method above.

So seemingly my aproach to calculate the blowup on affine charts and then patch the pieces together cannot applied here due to lack of knowledge of the ideal sheaf $\mathcal{I}$ associated to $X \subset \mathbb{P}^2$ even if I know the ideal sheaf $\mathcal{J}$ associated to the nine points as subscheme in $\mathbb{P}^2$ along which $X$ is blowed up. Indeed for the blowup constructions above I need to know the structure sheaf of $X$ and $\mathcal{J}$ as ideal sheaf with respect $X$, not $\mathbb{P}^2$.

Therefore I'm stuck at this point and my question is how to calculate the blowup "by hand" of this pencil along the nince points and if this strategy could be generalized to arbitrary pencils?

note I posted identical question a week ago on MSE without getting any resonance.