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Let $X$ be a geometrically irreducible, possibly singular quasi-projectiveprojective variety over an infinite field $k$. Assume that the dimension of $X$ is at least 2. Can there exist a hyperplane section of $X$ that is not geometrically connected?

Let $X$ be a geometrically irreducible, possibly singular quasi-projective variety over an infinite field $k$. Assume that the dimension of $X$ is at least 2. Can there exist a hyperplane section of $X$ that is not geometrically connected?

Let $X$ be a geometrically irreducible, possibly singular projective variety over an infinite field $k$. Assume that the dimension of $X$ is at least 2. Can there exist a hyperplane section of $X$ that is not geometrically connected?

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user145520
user145520

Bertini theorem for connectedness

Let $X$ be a geometrically irreducible, possibly singular quasi-projective variety over an infinite field $k$. Assume that the dimension of $X$ is at least 2. Can there exist a hyperplane section of $X$ that is not geometrically connected?