Ag.algebraic geometry – Reading geometric properties of an appropriate scheme from its refinement

Let $ k to be a field, $ X rightarrow mathrm {Spec} , k $ to be an appropriate morphism.

  • Yes $ X $ is geometrically reduced, so $ mathcal {O} _X (X) $ is the direct product of finely separable finite extensions of $ k.
  • Yes $ X $ is geometrically connected, then $ mathcal {O} _X (X) $ is an irreducible geometrically $ k-algebra.

Are the reverse statements true?