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Let $X$ be a smooth, proper scheme and $U\subset X$ be a normal and locally closed subscheme. Let $V\subset X$ be an open dense subscheme such that $U\subset V$ and $X\setminus V$ is a normal crossing divisor. What can I say about the boundary of the closure $\bar{U}$ of $U$ in $X$? Is $\bar{U}$ still normal?

Thx

1 Answers 1

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Non.

Let $X$ be the projective plane, $C$ a closed integral curve in $X$ with only one singular point $P$, let $V$ be the complement of a line $L$ passing through $P$ and $U=V\cap C$. Then $U$ is smooth, but $\overline{U}=C$ is not normal.