4
$\begingroup$

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.

$\endgroup$
3
  • 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
  • 3
    $\begingroup$ 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
5
$\begingroup$

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 $K$.

$\endgroup$
2
  • $\begingroup$ 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
  • 1
    $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.