Let $h \in \mathbb{C}[x]$ be separable of degree $d \geq 2$, in other words, there exist distinct $c_1,\ldots,c_d \in \mathbb{C}$ such that $h=(x-c_1)\cdots(x-c_d)$.
Let $I:=\langle h,y \rangle \subset \mathbb{C}[x,y]$, be the ideal of $\mathbb{C}[x,y]$ generated by $h$ and $y$. $I$ is a radical ideal and a finite intersection of the $d$ maximal ideals $\{\langle x-c_i,y \rangle\}_{1 \leq i \leq d}$.
Claim: There exist a finite number of ideals of $\mathbb{C}[x,y]$ containing $I$; the list is $\langle g,y \rangle$, where $g | h$.
For example, if $d=3$ and $c_1=\alpha, c_2=\beta, c_3=\gamma$, then the following ideals are containing $I= \langle (x-\alpha)(x-\beta)(x-\gamma),y\rangle$: $\mathbb{C}[x,y]$, $\langle x-\alpha, y\rangle$, $\langle x-\beta, y\rangle$, $\langle x-\gamma, y\rangle$, $\langle (x-\alpha)(x-\beta),y \rangle$, $\langle (x-\alpha)(x-\gamma), y\rangle$,$\langle (x-\beta)(x-\gamma), y\rangle$, $I$.
Question: Is my claim true? Am I missing something?
Remarks:
(1) I have used the assumption that $h$ is separable, so, for example, $\langle (x-\alpha)(x-\beta),x-\gamma,y \rangle$ becomes $\mathbb{C}[x,y]$, because $(x-\alpha)(x-\beta),x-\gamma$ generate $1$, see this.
Notice that, $\langle (x-\alpha)(x-\beta),(x-\alpha)(x-\gamma),y \rangle$ is actually $\langle x-\alpha, y \rangle$ which is already listed above; indeed, there exist $U,V \in \mathbb{C}[x]$ such that $U(x-\beta)+V(x-\gamma)=1$, hence, $U(x-\alpha)(x-\beta)+V(x-\alpha)(x-\gamma)=(x-\alpha)[U(x-\beta)+V(x-\gamma)]=x-\alpha$.
(2) Of course, we can replace the generators of the ideals in the list, for example, instead of $\langle x-\alpha,y \rangle$, we can write $\langle (x-\alpha)+y^5,y \rangle$.
(3) Is it just because $I= \cap_{i=1}^{d} \langle x-c_i,y \rangle$ and such an intersection is contained only in partial intersections. Indeed, Let $\mathbb{C}[x,y] \supsetneq J \supseteq I$, $J$ arbitrary proper ideal containing $I$. There exists a maximal ideal $M$ of $\mathbb{C}[x,y]$ containing $J$: $M \supseteq J \supseteq I$, so, $M \supseteq I= \prod_{i=1}^{d} \langle x-c_i,y \rangle$. Therefore, $M = \langle x-c_j,y \rangle$, for some $1 \leq j \leq d$.
(4) Related question: i and ii.
Any comments are welcome, thank you!