Suppose $R=\mathbb{C}[x_1,\ldots, x_n]$ is a polynomial ring with $I$ being an ideal of $R$. Let $I'$ be an ideal of $R[t]$.
If $R[t]/I'$ is flat as a $\mathbb{C}[t]$-module and over $0$, $\dfrac{R[t]}{I'}\otimes_{\mathbb{C}[t]}\dfrac{\mathbb{C}[t]}{(t)}$ is reduced, then over any other point, is $ \dfrac{R[t]}{I'}\otimes_{\mathbb{C}[t]}\dfrac{\mathbb{C}[t]}{\langle t-c\rangle} \cong \dfrac{R}{I} $ for $c\not=0$ reduced with $ dim \left( \dfrac{R[t]}{I'}\otimes_{\mathbb{C}[t]}\dfrac{\mathbb{C}[t]}{(t)}\right) = dim\left(\dfrac{R[t]}{I'}\otimes_{\mathbb{C}[t]} \dfrac{\mathbb{C}[t]}{\langle t-c\rangle} \right)? $
$ $ Here is one example (which may possibly satisfy the above two conditions) that I would like to calculate as a hands-on concrete exercise.
$I' = \langle x_1-x_2+x_3, x_1 y_1 + z_1 y_2, x_3 y_3-z_2 y_2, x_2 y_2 +(1-t)z_2^2\rangle $ in $R[t]=\mathbb{C}[x_1, x_2, x_3, y_1, y_2, y_3,z_1, z_2][t]$.