Field of moduli of certain generalized Fermat Curves

Abstract

Let K < F be a field extension and X ⊂ Pⁿ, a projective algebraic variety defined over F. A field of definition for X is a field N so that K < N < F and there exists a projective algebraic variety Y defined over N which is birational equivalent (or conformal equivalent) to X. Let Aut(F/K) be the automorphism group of the field extension K < F. Applying σ ∈ Aut(F/K) to every coefficient of the polynomial which defines X is an action group. This action provides us a new projective algebraic variety, X^σ, which in general is not equivalent with X. Let us consider F_K = {σ ∈ Aut(F/K) : X ≅ X^σ} the set of all σ’s to make X and X^σ equivalent in some sense (birationally equivalent, conformally equivalent ... ). The fixed field of F_K is called the field of Moduli of X. This is denoted by M_{F/K}(X). From the definition, it is easy to see that K < M_{F/K}(X) < F, so the question is: Could M_{F/K}(X) be a field of definition for X? To answer that question, we have Weil's theorem which gives necessary and sufficient conditions to determine when a field of moduli is a field of definition. A pair (S, H) is called a generalized Fermat pair of type (k, n) if S is a closed Riemann surface, H ≅ Z_k^n is its automorphism group, and the quotient S/H is a Riemann's orbifold with n + 1 cone points and signature (0; k, … , k). S is called generalized Fermat curve of type (k, n) and H generalized Fermat group of type (k, n). Using a consequence of Riemann–Roch's theorem, we can relate S with a projective algebraic variety. In this thesis we are going to prove that the field of moduli of a generalized Fermat curve of type (k, 4) is also a field of definition. To achieve this we will use a modified version of Dèbes – Emsalem's theorem, which is an application of Weil's theorem.