Lemma 115.11.1. Let $(R, \mathfrak m, \kappa )$ be a local ring. Let $X \subset \mathbf{P}^ n_ R$ be a closed subscheme. Assume that $R = \Gamma (X, \mathcal{O}_ X)$. Then the special fibre $X_ k$ is geometrically connected.

**Proof.**
This is a special case of More on Morphisms, Theorem 37.53.5.
$\square$

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