Plane model-fields of definition, fields of definition, and the field of moduli for smooth plane curves

Let C / k ‾ be a smooth plane curve defined over k ‾ , a fixed algebraic closure of a perfect field k. We call a subfield k ′ ⊆ k ‾ a plane model-field of definition for C if C descends to k ′ as a smooth plane curve over k ′ , that is if there exists a smooth curve C ′ / k ′ defined over k ′ which is k ′ -isomorphic to a non-singular plane model F ( X , Y , Z ) = 0 with coefficients in k ′ , and such that C ′ ⊗ k ′ k ‾ and C are isomorphic. In this paper, we provide (explicit) families of smooth plane curves for which the three fields types; the field of moduli, fields of definition, and plane-models fields of definition are pairwise different.