Example 33.7.2. Let $k = \mathbf{Q}$. The scheme $X = \mathop{\mathrm{Spec}}(\mathbf{Q}(i))$ is a variety over $\mathop{\mathrm{Spec}}(\mathbf{Q})$. But the base change $X_{\mathbf{C}}$ is the spectrum of $\mathbf{C} \otimes _{\mathbf{Q}} \mathbf{Q}(i) \cong \mathbf{C} \times \mathbf{C}$ which is the disjoint union of two copies of $\mathop{\mathrm{Spec}}(\mathbf{C})$. So in fact, this is an example of a non-geometrically connected variety.

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