An algebraic variety $X$ over a field $K$ is a $K$-scheme integral separated and of finite type over $K$.
Definition: If $L$ is a subfield of $K$ we say that $X$ is defined over $L$ if there exists a variety $X_L$ over $L$ such that
$$X\cong X_L\times_{\operatorname{Spec} L}\operatorname{Spec} K$$
where the morphism $\operatorname{Spec} K\longrightarrow \operatorname{Spec} L$ is that induced by the immersion $\iota$ of $L$ in $K$, whereas the morphism $X_L\longrightarrow\operatorname{Spec} L$ is the structural morphism of the variety $X_L$. If $X$ is defined over $L$, then $L$ is called a field of definiton of $X$.
The above definition is standard in the framework of schemes and it is coherent with Mumford's notations. Now consider a finite morphism $t:X\longrightarrow \mathbb P^1_K$ where $X$ is a curve over $K$ (variety of dimension $1$), then Bernhard Köck in his article "Belyi's theorem revisited" cites the field of definition of $t$ without defining it. I don't understand what this object can be! Do you have any idea? Is there a standard definition for "a field of definition of a morphism between curves"?
Thanks in advance.
To a morphism of varieties $f:X\rightarrow Y$ you can attach the subvariety $$ Z_f=\{(x,f(x)\}_{x\in X}\subseteq X\times Y. $$ The field of definition of $f$ is the field of definition of $Z_f$.
In concrete terms, Zariski-locally $f$ is described by polynomial functions. The morphism is defined over the field $K$ when those polynomials have coefficients in $K$.
Also, a morphism defined over $K$ will send $K$-rational points of $X$ to $K$-rational points of $Y$, whereas this is not generally true even if the varieties are defined over $K$.