Theorem: (Weil’s theorem)
Let < be a finite Galois extension,
a projective algebraic variety defined over and . Suppose that for each exists a birational application defined over such that, for each pair , holds, that is the following diagram commutes

Then:
1.- There exists a projective algebraic variety , defined over , and there exists a birational application defined over such that, for every it holds the equality . Moreover, if every is biregular, then we can assume to be a biregular isomorphism.
2.-  If the pair is another solution for the above problem, then exists a birational isomorphism defined over such that,

PROOF: of Weils’ theorem

Now, if is a non-singularprojective algebraic curve of genus defined over a field , then its automorphism group is finite. We can consider the Galois covering . And in this case,

And as consequence we have el teorema de Dèbes and Ensalem [P. Dèbes, M. Emsalem, On fields of Moduli. J. of Algebra. 1999]

Theorem: (Dèbes-Emsalem)
Let < be a finite Galois extension, a non- singular projective algebraic curve of genus defined over , and . Let be the birational automorphism group of defined over . Let be the Galois covering with cover group . Then:
1.- Exist a non-singular projective algebraic curve , defines over  (called canonic model of ), and an isomorphism defined over , such that if , then .
2.- Moreover, if we can fiend a point in , where is the set of all critical values of , which are -rational, then is a definition field of .

PROOF: of Debes-Emsalem’s Theorem

Let < be an extension fields, let be projective varieties defined over and let be an algebraic morphism of finite degree over . We also assume that is defined over .

Applying to , we obtain a new morphism . Since is defined over , this implies that .

It said that the morphism is equivalent to , which is denoted, if there exists a birational isomorphism , such that the following diagram commutes

that is, .

Definition: (Field of Moduli for morphism)
Let < be an extension field, let be projective algebraic varieties defined over and let be an algebraic morphism of finite degree over . We also assume that is defined over . Consider the following subgroup of

We define the field of Moduli of , respect to the extension fields <, as the fixed field of the subgroup , i.e.

Remark: Notice form the definition that   , but in general the might be different.

If the morphism is a Galois morphism, that is is defined by the action of a finite subgroup of , then for each   we can see the group <.

Note that, if , than it is not totally clear the existence of an isomorphism , such that . But, if the group is unique (in some sense), for example when , then it is true and we have the next result.

Theorem: Let be a non-singular projective algebraic variety,   <  a finite group of birational automorphism of and an algebraic morphism by the action of so that is defined over . If the group is unique in , then

We are going to start with a projective variety .  If    is an homogeneous polynomial of degree , then we have the associated projective algebraic curve

If in non- singular (i.e their partial derivates do not vanish simultaneously at any point of ) then we can endow to   with a compact Riemann surface structure.

So, here is our first definition

Definition: (Fields of definition)

Let < be an extension field and     let be a projective algebraic variety. A field of definition for is any field , such that <<, so that there exists a projective algebraic variety  birational equivalent to ().

Now, consider < an extension field, Aut() its automorphism group and    the polynomial ring of with unknown variables .

If

and ,   Aut()   then we have the natural action of on given as 

This provides a natural action of Aut() over the ring .

If we consider    a projective algebraic variety defined by

then every Aut() provides a new projective algebraic variety defined by

Theorem: Let < be an extension field and let , be birational equivalent projective algebraic varieties. Then, for every Aut() it holds that and are also birational equivalent.

Given , a natural question is how different can be and , when run through Aut().
If we denote by the set of birational equivalent classes of projective algebraic varieties defined over , then we have that the action described above induces in a natural way the following action

The orbit of [] is conformed by all the classes of algebraic varieties [] which are birational equivalent to , for all Aut(), that is,

The stabilizer of [], with respect to the above action, is given by

A field of Moduli is the smallest field where a Riemann surface can be defined, its definition is

Definition: (Field of Moduli)

Let < be an extension field an let be a projective algebraic variety. The field of Moduli for , with respect to the extension field < , is

Remark:

Let < be an extension fields and let be a projective algebraic variety.

1.- By the above definition, it easy to see that < < .
2.- If < is a general Galois extension and is defined over , then = .

Indeed: Since is defined over , then for every Gal(), .

Thereby,

3.- Let be a field such that, < < . Since Aut() is a subgroup of Aut(), we have that is a subfield of . But in general,

Now we are going to see how field of Moduli are related ti the fields of definition.

Theorem: Let < be a general Galois extension and let be a projective algebraic variety. Then, every field of definition of contains the field of Moduli of .

PROOF: Let < < be an extension fields, where is a fields of definition for . We can assume that is defined by homogeneous polynomials with coefficients in .
If we take Aut() < Aut(), then , this means , which implies Aut() < . Therefore,

Another theorem that could be useful
Theorem: Let be an algebraically closed field, let be the prime field of and let be a projective algebraic variety. Then, there exists a field of definition for which is a finite extension of the field of Moduli .
PROOF: See H. Hammer – F. Herrlich, “A remark on the Moduli Field of a Curve”  Arch. Math. 81. 2003.

Note that the above theorem works for projective algebraic varieties defined over and taking .

To end this first post, I will state a theorem whose proof is not hard, but you must be careful with writing.

Theorem: Let < be an extension field, where is algebraically closed, and let be a projective algebraic variety. Let be the algebraic closure of the field of Moduli in . Then,