Let $S:= H_1\bigcap H_2\bigcap \cdots \bigcap H_N \subset\mathbb{P} _{\mathbb{C}}^{N+2}$ be a complete intersection surface, where each $H_i$ is a hypersurface defined by a homogeneous equation $f_i$.
If $x\in S$ is a singular point of $H_i$ for some $i$, then $x$ is a singular point of $S$.
My question is: Is it possible that there is a point $x\in S$ such that each $H_i$ is smooth at $x$ and $S$ is singular at $x$? Can you give a reason for your answer?
this is not a research level question. a point on an intersection is singular if either it is singular on some one of the intersecting varieties or if those varieties are tangent at some point.