Let $A$ be an integral Noetherian $\mathbb{C}$-algebra. Denote by $\mathbb{P}^3_A:=\mathbb{P}^3_{\mathbb{C}} \times_{\mathbb{C}} \mathrm{Spec}(A)$. Let $X,Y$ be closed local complete intersection subschemes (not necessarily reduced or irreducible) in $\mathbb{P}^3_A$ of pure dimension $\dim A+1$ flat over $\mathrm{Spec}(A)$. Assume that $X \subset Y$. Denote by $\mathcal{I}$ the ideal sheaf obtained as the ideal quotient $[\mathcal{I}_Y:\mathcal{I}_X]$.
Assume further that $X$ is reduced and the reduced scheme associated to $Y$ is $X$. Then,
1) Is the scheme defined by $\mathcal{I}$ flat over $\mathrm{Spec}(A)$?
2) Does the ideal quotient agree with specialization i.e., for any maximal ideal $m$ of $A$, is $[\mathcal{I}_Y:\mathcal{I}_X] \otimes_A A/m \cong [\mathcal{I}_Y \otimes_A A/m:\mathcal{I}_X \otimes_A A/m]$?
Any idea/reference in this direction will be most appreciated.
