Let $X$ be a reduced $k$-scheme ($k$ an algebraically closed field) and consider a $k$-morphism, $$X \to \operatorname{Spec}k[t]$$ such that the associated $k$-algebra map, $$k[t] \to \Gamma(X,\mathcal{O}_X)$$ is injective. Under what conditions I can assure that this morphism will be dominant? If the morphism is dominant then it follows that the generic point of the affine line must lie in the set theoretical image of $X$?
Schemes over $k$ and injectivity.
1
$\begingroup$
algebraic-geometry
-
2What is your definition of dominant? If it is "dense image", then the injectivity implies dominance. But the this doesn't necessarily imply the generic point is in the set-theoretic image unless $X$ is of finite type. E.g. if $X$ is the disjoint union of closed points labeled by points $a \in k$, and if $X \to $ Spec $k[t]$ is the obvious map identifying $X$ with the closed points of the target, then the map on global sections in injective. – 2017-01-20