Let $C_1, C_2$ be two distinct cubic curves in $\mathbb{P^2}$ with $F_1, F_2 \in \mathbb{C}[X,Y,Z]$ the homogeneous cubic polynomials generating the vanishing ideals of $C_1$, respectively $C_2$. 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 $\mathbb{P}^1$.
My question is if it possible to describe $X$ explictly by vanishing polynomials, respectively the associated ideal sheaf generated by these?
Say for example that subschemes $X_1,..., X_n \subset \mathbb{P^2}$ are defined by vanishing polynomials $F_1,..., F_n \in \mathbb{C}[X,Y,Z]$. Then the scheme $X_1 \cup... \cup X_n \subset \mathbb{P^2}$ is associated with polynomial $F_1 \cdot F_2 \cdot ... \cdot F_n$. (More generally if $X_1,..., X_k \subset \mathbb{P^2}$ where each $X_i$ is defined by vanishing polynomials $F_{i,1}, F_{i,2},..., F_{i, m(i)}$, then the subscheme $X_1 \cup... \cup X_k\subset \mathbb{P^n}$ is associated to the vanishing polynomials $ F_{1, i_1} \cdot F_{2, i_2} \cdot ... \cdot F_{k, i_k}$.)
The problem with pencil $X$ is that it is not a finite union or subschemes, therefore the product of all $\lambda F_1 + \mu F_2$ is meaningless. Are there techniques known to determine the vanishing ideal of $X$?
This problem arise naturally when one wants to blowup $X$ in certain points, since it requires knowledge of the structure sheaf of $X$. see also this question for details
If you consider $X$ as a subset of $\mathbb{P}^2$ then $X = \mathbb{P}^2$ (because for any point of $\mathbb{P}^2$ there is a pair $(\lambda,\mu)$ such that $\lambda F_1 + \mu F_2$ vanishes at that point).