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.

  • 2
    $\begingroup$ Presumably it's exactly what you've written. Namely, there is a scheme $X_L$ and a morphism $t:X_L\to\mathbb{P}^1_L$ such that base-changing to $K$ gives you $t$. $\endgroup$ Mar 20 '14 at 7:09
  • $\begingroup$ I don't understand very well the details of the definition. Can you write it formally? Many thanks. $\endgroup$
    – Dubious
    Mar 20 '14 at 8:20
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    $\begingroup$ I'm just saying that there is some scheme $X_L$ over $L$ such that $X_L\times\text{Spec}(k)=X$, and a morphism $t_L:X_L\to\mathbb{P}^1_L$ such that the map $t_L\times \text{id}_K:X_L\times\text{Spec}(K)\to \mathbb{P}^1_L\times\text{Spec}(K)$ is just your $t$ (note that $\mathbb{P}^1_L\times\text{Spec}(K)=\mathbb{P}^1_K$). $\endgroup$ Mar 20 '14 at 8:25

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 $K$.

  • $\begingroup$ why a morphism defined over $K$ will send $K$-rational points of $X$ to $K$-rational points of $Y$? How can I see it? $\endgroup$
    – Dubious
    Mar 24 '14 at 12:33
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    $\begingroup$ Assume for the sake of simplicity that your varieties $X$ and $Y$ are projective. Then the $K$-rational points of $X$ and $Y$ are those admitting homogeneous coordinates in $K$. The morphism $f$, being itself defined over $K$, admits a description in terms of homogeneous polynomials with coefficients in $K$. But when you evaluate polynomials with coefficients in $K$ on a tuple of elements of $K$ the result is a tuple of elements of $K$. $\endgroup$ Mar 24 '14 at 18:00

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