Suppose $A$ is an abelian variety over an algebraically closed field $k$. I'm confused about the notion of translates of abelian subvarieties. I looked over a few related papers, but I couldn't find a precise algebro-geometric definition. People simply used $x+B$, and I assume they are interested in functor of points. In my opinion, given a point $x:\mathrm{Spec}k'\rightarrow A$, a translate should live in the base change $A\_{k'}$. Namely there is an abelian subvariety $B$ of $A\_{k'}$ such that $x+B$ is our translate. 


If $x$ is a closed point of $A$ and $B\subset A$, then $x+B$ would be a closed subvariety of $A$.


If $x$ is not closed, a translate through $x$ is really something lives in a base change of $A$ (so that $x$ becomes a closed point in the base change). 


I'm wondering about the following question:


Mordell exceptional locus of a closed subvariety $X\subset A$ is defined as the union of images of positive-dimensional translates inside $X$. Do we need to consider the translates through non closed points? Namely, if we take the union of translates through closed points, will that give us the same thing?



Translates of abelian subvarieties
Usually, when one speaks of "translate of an abelian subvariety" it really is in the sense "translate of an abelian subvariety by a closed point". 


To answer your question: Let $X\subset A$ be a closed subvariety of an abelian variety $A$ over $k$. Assume that $k$ is algebraically closed of characteristic zero. Let $Sp(X)$ be the union of positive-dimensional translates (by closed points of $A$) of abelian subvarieties contained in $X$. Kawamata proved that $Sp(X)$ is closed in $X$; see Thm. 4 in Y. Kawamata, On Bloch’s conjecture, Invent. Math. 57 (1980), 97-100. 


**Side remark.** Ueno proved that $Sp(X) \neq X$ if and only if $X$ is of general type. Similar statements are true for closed subvarieties of semi-abelian varieties in positive characteristic; see Abramovich numdam.org/item/CM\_1994\_\_90\_1\_37\_0 


Let $L/k$ be an extension of algebraically closed fields. Then $Sp(X)\_L = Sp(X\_L)$, where $Sp(X\_L)$ is the union of 
the positive-dimensional translates of abelian subvarieties of $A\_L$ contained in $X\_L$. Thus, the "special" locus of $X$ behaves well with respect to field extensions. Let me explain how to prove this in a more general context.


Let $\Delta^{gr}\_X$ be the groupless-exceptional locus. That is, $\Delta^{gr}\_X$ is the Zariski closure of the union of the images of non-constant morphisms $U\to X$, where $U$ is a dense open subset of a connected finite type group scheme $G$ over $k$ such that $\mathrm{codim}\_G(G\setminus U)\geq 2$. 
 Then, $\Delta^{gr}\_X =Sp(X)$; see Theorem 13.1 in <https://arxiv.org/pdf/2002.11981.pdf> .
To prove this equality of sets, use the following three facts:



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> > 1) every rational map $B\dashrightarrow X$ with $B$ an abelian variety extends to a morphism $B\to X$ (use that $X$ has no rational curves). 
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> > 2) if $G$ is a connected linear algebraic group, then every rational map $G\dashrightarrow X $ is constant. This is because linear algebraic groups are covered by (non-compact) rational curves.
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> > 3) The image of a morphism of abelian varieties $B\to A$ is the translate of a abelian subvariety of $A$. 
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In the absence of an ambient abelian variety, the general statement you are looking for is the following:



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> > **Proposition 3.7 in <https://arxiv.org/abs/1909.12187>**
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> > *Let $L/k$ be an extension of algebraically closed fields of characteristic zero. Let X be a proper variety over $k$. Then $(\Delta\_X^{gr})\_L = \Delta^{gr}\_{X\_L}$.*
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