Let $X$ be a regular integral noetherian scheme of dimension 2 and let $D$ be a simple normal crossings divisor in $X$. 


EDIT: Let $U = X-D$.


Consider a finite etale morphism $V\longrightarrow U$ with $V$ connected. Let $\pi:Y\longrightarrow X$ be the normalization of $X$ in the function field of $V$. So $Y$ is a $2$-dimensional normal noetherian scheme and $\pi$ is finite.


One can show that $\pi$ is surjective, flat and that $Y$ is CM.


Q: Is $\pi$ a local complete intersection?



Is this finite surjective flat morphism of 2 dimensional schemes a local complete intersection
Let $k$ be a field of characteristic prime to $n$ and $\zeta\in k$ a primitive $n$-th root of unity. Let $a\in \mathbb{Z}$ and let $Y$ be the quotient of $\mathop{\rm{Spec}}k[u,v]$ by the automorphism $(u,v)\mapsto (\zeta u,\zeta^a v)$. 


Then $Y$ is a tame cyclic cover of $X=\mathop{\rm{Spec}}k[u^n,v^n]$ etale away from $D=\{uv=0\}$. Typically $Y$ is not an lci. For example, if $(n,a)=(3,1)$ then 
$$Y=\mathop{\rm{Spec}} k[u^3,v^3,uv^2,u^2v]=\mathop{\rm{Spec}}k[x,y,z,t]/(xy-zt,xz-t^2,yt-z^2).$$ 
If you allow wild ramification I suspect you can produce an example in which $D$ is even a smooth divisor.


