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?

## 1 Answer

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Kevin Casto's comment appears to do the trick. But in case a reference would be helpful, here is a result from this paper by Martinelli, Naranjo, and Pirola:

Theorem 1.1:Let $k$ be an algebraically closed field, $X$ an irreducible projective variety over $k$, and $f:X\to\mathbb{P}_k^n$ a morphism. If $r\geq n+1-\text{dim}\,f(X)$, then for any $r$-dimensional linear subvariety $L\subseteq \mathbb{P}_k^n$, $f^{-1}(L)$ is connected.

In particular, if $\text{dim}\,f(X)\geq 2$, every hyperplane section is geometrically connected.

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