Let $f: X \to Y$ be a morphism between noetherian integral schemes. Suppose $f$ is finite flat of degree two. Then is there an involution $\sigma: X \to X$ over $Y$, which gives the non-trivial Galois involution at the generic point?

Sure. Consider what happens at the level of rings $B \to A$ and attempt to define the involution there.
– Abhinav KumarJul 7 '13 at 23:48

6

You are implicitly assuming that $f$ is generically etale. In characteristic 2, there are finite flat morphisms of degree 2 that are purely inseparable.
– Jason StarrJul 8 '13 at 0:40

6

Since it is finite flat of degree 2, Zariski-locally on the base it is ${\rm{Spec}}(A)\rightarrow{\rm{Spec}}(B)$ where 1 is part of a $B$-basis $\{1,a\}$ of $A$. If $T^2-uT+v$ is the characteristic polynomial of $a$-multiplication on $A$ then $B[T]/(T^2-uT+v)\rightarrow A$ via $X\mapsto a$ is an isomorphism. Then $T \mapsto u-T$ is an $A$-automorphism that does the job when $T^2-uT+v$ is separable over Frac($B$).
– user61789Jul 8 '13 at 1:39

Good point, Jason :-)
– Abhinav KumarJul 8 '13 at 13:36

If $X$ and $Y$ are also regular then any finite generically etale morphism of degree $2$ between them is flat and $X$ is the normalization of $Y$ in the function field of $X$ (by Zariski's main theorem). It thus naturally carries an extension of the generic involution by the construction of the normalization.
– Damian RösslerJul 8 '13 at 22:21