I want to follow this construction of a normal model of a curve:
Let $p\neq 2,3$ and $Y\to \mathbb P¹$ be a smooth projective curve over $\mathbb Q_p$ with function field $L/\mathbb Q_p(x)$ e.g. $L=\mathbb Q_p(x)[y]/(y²-(x-p)(x+p)(x-2))$. We obtain a model $\mathcal Y$ of $Y$ by normalizing $\mathbb Z_p[x]$ in $L$.
Now I see that $Y$ (the generic fiber of $\mathcal Y$) is an elliptic curve and that $\mathcal Y\otimes \mathbb F_p$ (the special fiber) is a singular curve.
My question is: Does the normalization (integral closure of $A=\mathbb Z_p[x]$ in $L$) fix this problem of the singular special fiber? And how does the normalization look like in the above example $L=\mathbb Q_p(x)[y]/(y²-(x-p)(x+p)(x-2))$?
Kind regards
