Suppose $Q_1, Q_2\in \mathbb{C}[X_0,\dots,X_n]$ are irreducible homogeneous quadratic polynomials such that $V(Q_1, Q_2)$ is an irreducible projective variety of degree two and codimension two in $\mathbb{P}^n$ (note the projective subscheme defined by $Q_1$ and $Q_2$ has degree 4 and hence must be nonreduced). Let $I$ be the homogeneous height-one prime ideal of $\mathbb{C}[X_0,\dots,X_n]/(Q_1)$ that defines $V(Q_1, Q_2)$ as a subvariety of $V(Q_1)$. Is $I$ always principal, i.e., generated by a linear polynomial?
Edit: feel free to assume there is another homogeneous quadratic polynomial $Q_3$ such that $Q_3\not\in (Q_1,Q_2)$ and $Q_3\bmod (Q_1)\in I$.
If $\mathbb{C}[X_0,\dots, X_n]/(Q_1)$ is a UFD then $I$ is obviously principal, as height-one prime ideals of UFDs are pricinpal. In other cases I don't know how to proceed.