Geometric properties of fibres

For a morphism $X\to S$ and $s\in S$, call $X_s$ geometrically reduced, geometrically irreducible, geometrically integral, or geometrically connected when the chosen algebraic-closure fibre of Geometric fibres and geometric points has the corresponding property. Reduced means all local rings have no nonzero nilpotents, equivalently its reduction from The reduction of a scheme is itself. Irreducible here requires a nonempty space not expressible as a union of two proper closed subsets. Integral means reduced and irreducible, with nonemptiness, as in Integral schemes. Connected means no separation into two nonempty disjoint open subsets, as in Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets. Thus an empty geometric fibre is reduced and connected, but neither irreducible nor integral.

Under Choice, Independence of the chosen algebraic closure makes these tests independent of the choice of algebraic closure. This page uses these closure tests as its convention; an equivalence with tests over every extension field is not needed in the proofs here.