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Let $f: X \to Y$ be a finite map of projective varieties.

I'm trying to understand when and how often should i expect $f$ to be a quotient map by a finite group acting on $X$. Even more strictly let $G=Aut(k(X)/k(Y))$.

1. When is there a well defined action of $G$ on $X$?

2. When is $f$ isomorphic to the quotient map by an action $X \to X^G$?

If $X$ and $Y$ are smooth projective curves then I think (2) always holds (although $G$ may be different) but I haven't been able to formalize this. Are there any useful conditions which ensure that (2) holds?

Let $f: X \to Y$ be a finite map of projective varieties.

I'm trying to understand when and how often should i expect $f$ to be a quotient map by a finite group acting on $X$. Even more strictly let $G=Aut(X/Y)$.

1. When is $f$ isomorphic to the quotient map by an action $X \to X^G$?

If $X$ and $Y$ are smooth projective curves then I think this always holds but I haven't been able to formalize this. Are there any useful conditions which ensure that (1) holds?

EDIT: I deleted the irrelevant part of the question. I state it as a different question: When is $Aut(X/Y)=Aut(k(X)/k(Y))$?

Let $f: X \to Y$ be a finite map of projective varieties.

I'm trying to understand when and how often should i expect $f$ to be a quotient map by a finite group acting on $X$. Even more strictly let $G=Gal(\widetilde{k(X)}/k(Y))$ where $\widetilde{k(X)}$ is the Galois closure.

1. When is there a well defined action of $G$ on $X$? (Generically such an action always exists).

2. When is $f$ isomorphic to the quotient map by the Galois action $X \to X^G$?

If $X$ and $Y$ are smooth projective curves then I think this is always the case but I haven't been able to formalize this. Are there any useful conditions which ensure that (2) holds?

Let $f: X \to Y$ be a finite map of projective varieties.

I'm trying to understand when and how often should i expect $f$ to be a quotient map by a finite group acting on $X$. Even more strictly let $G=Aut(k(X)/k(Y))$.

1. When is there a well defined action of $G$ on $X$?

2. When is $f$ isomorphic to the quotient map by an action $X \to X^G$?

If $X$ and $Y$ are smooth projective curves then I think (2) always holds (although $G$ may be different) but I haven't been able to formalize this. Are there any useful conditions which ensure that (2) holds?

Let $f: X \to Y$ be a finite map of projective varieties.

I'm trying to understand when and how often should i expect $f$ to be a quotient map by a finite group acting on $X$. Even more strictly let $G=Gal(\widetilde{k(X)}/k(Y))$ where $\widetilde{k(X)}$ is the Galois closure.

1. When is there a well defined action of $G$ on $X$? (Generically such an action always exists).

2. When is $f$ isomorphic to the quotient map by the Galois action $X \to X^G$?

If $X$ and $Y$ are curves then I think this is always the case but I haven't been able to formalize this. Are there any useful conditions which ensure that (2) holds?

Let $f: X \to Y$ be a finite map of projective varieties.

I'm trying to understand when and how often should i expect $f$ to be a quotient map by a finite group acting on $X$. Even more strictly let $G=Gal(\widetilde{k(X)}/k(Y))$ where $\widetilde{k(X)}$ is the Galois closure.

1. When is there a well defined action of $G$ on $X$? (Generically such an action always exists).

2. When is $f$ isomorphic to the quotient map by the Galois action $X \to X^G$?

If $X$ and $Y$ are smooth projective curves then I think this is always the case but I haven't been able to formalize this. Are there any useful conditions which ensure that (2) holds?