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.
Best Answer
If the two cubics $C_1=V(F_1)$ and $C_2=V(F_2)$ do not share a common component, then the ideal $I = (F_1,F_2)$ defines a $0$-dimensional subscheme $V(I)\subset \mathbb{P}^2$ length 9, which is a complete intersection of codimension 2.
In the fancy language above, the graded ring $\bigoplus_{n\geq0}I^n$ is generated as an algebra over $\mathcal{O}_{\mathbb{P}^2}$ by two elements $\xi_1,\xi_2$ in degree $1$, corresponding to the two equations $F_1,F_2$ generating $I$. They satisfy one relation $\xi_1F_2-\xi_2F_1=0$ corresponding to the syzyzy that holds between them.
Thus the blowup of $Z:=V(I)\subset \mathbb P^2$ is explicitly given by the projective variety $$ \overline X = V(\xi_1F_2 - \xi_2F_1) \subset \mathbb P^1_{\xi_1,\xi_2}\times \mathbb P^2_{x,y,z}$$ and the morphism $\overline \pi\colon \overline X\to \mathbb{P}^1$ that you want to consider is simply the projection onto the first factor. (The blowup itself is given by projection onto the second factor.)
Unfortunately, if $Z$ is not reduced then this blowup $\overline X$ is going to be rather singular, so at this point you probably want to replace $\overline X$ with its minimal resolution $\mu\colon X\to \overline X$ and consider the induced morphism $\pi\colon X\to \mathbb P^1$.
I want to note that this construction is probably not exactly the approach Miranda had in mind. Judging from what he wrote he considers an iterative construction:
If you think hard enough about it, then I think it will follow from (a) the universal property of blowing up, and (b) the existence of minimal resolutions of surfaces, that this construction gives the same surface as $X$ above. The advantage of the first approach is it is obvious how to get the morphism to $\mathbb P^1$. The advantage of the second approach is that it gives a smooth surface directly, without having to go through some desingularisation process at the end.
Lastly, by some abstract nonsense there should exist some sheaf of ideals $\mathcal J$ on $\mathbb P^2$ such that the morphism $\pi\colon X\to \mathbb{P}^2$ is given by blowing up $\mathcal J$. Perhaps you had in mind to construct $\pi\colon X\to \mathbb{P}^2$ all in one go by working out how to write down this $\mathcal J$. I would think this is highly unlikely to be a pleasant calculation.