Answer by grghxy for Field of definition of dominant morphisms
The general technique goes as follows. Let $\{A_i\}$ be a directed system of rings with limit $A$; e.g., $A=k$ and $\{A_i\}$ the set of finitely generated $k_0$-subalgebras of $k$. Let $X$ and $Y$ be...
View ArticleAnswer by Will Sawin for Field of definition of dominant morphisms
The same trick, spreading out, should work for all of these. Write down all the equations defining your morphism. Look at all the coefficients. Take the algebra generated by these over $k_0$. Each...
View ArticleField of definition of dominant morphisms
Let $k$ be an algebraically closed field and $k_0$ a sub-field. Let $X,Y$ be two projective varieties defined over $k_0$. Suppose that that there exists a dominant morphism $f$ between $X_k=X\otimes k$...
View Article
