Suppose $X$ is a smooth projective surface over complex numbers in $mathbb{P}^n$. Let $text{Spec}(R)$ be an affine open set of $X$. Then $R$ is a finitely generated $mathbb{C}$ algebra of dimension $2$. Let $Z:= {p_1, …p_k}$ be a set of distinct closed points in $text{Spec}(R)$. Since $X$ is smooth, there are regular parameters $x_i, y_i$ such that the maximal ideal $m_{p_i}$ is generated by $x_i$ and $y_i$. Let $I$ be ideal defined by $m^{‘}_{p_1}m^{‘}_{p_2}…m^{‘}_{p_k}$, where $m^{‘}_{p_i} = langle x_i^2, y_irangle$. Let $Z^{‘}$ be the subscheme defined by $I$.

What is the geometric interpretation of $Z^{‘}$ in terms of $Z$?

Is $Z^{‘}$ locally complete intersection?

Suppose $Y in H^0(I_{Z^{‘}}(d))$ is a degree $d$ hypersurface section. Then can we say that $Y$ is singular along $Z$ ?