# Translates of abelian subvarieties

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?

• 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 over an abelian variety $A$ over $k$. Assume $k$ 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 $X$. 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 ... Mar 31, 2020 at 17:09
• ...the positive-dimensional translates of abelian subvarieties by closed points of $A_L$ contained in $X_L$. Thus, the Mordell exceptional locus behaves well with respect to field extensions. You can prove this using spreading out and specialization arguments; see also Proposition 3.7 in arxiv.org/abs/1909.12187 Mar 31, 2020 at 17:09
• @AriyanJavanpeykar: That's really helpful! Thanks a lot! So I assume if we want an argument as Proposition 3.7, we would need to know $Sp(X_L)$ is closed？Do you think there is a way to do it without assuming closedness of the set (so $Sp(X)_L$ just means the inverse image of $Sp(X)$ in $X_L$)? Mar 31, 2020 at 23:48
• @AriyanJavanpeykar: Also, if you have a nontrivial map from a group variety to $X$, would it factor through $Sp(X)$? Apr 1, 2020 at 3:48
• The fact that $Sp(X)$ is closed in $X$ was proven first by Kawamata; see Thm. 4 in Y. Kawamata, On Bloch’s conjecture, Invent. Math. 57 (1980), 97-100. Ueno subsequently 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; see Abramovich numdam.org/item/CM_1994__90_1_37_0 ...cont'd below... Apr 1, 2020 at 8:16

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:

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).

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.

3) The image of a morphism of abelian varieties $$B\to A$$ is the translate of a abelian subvariety of $$A$$.

In the absence of an ambient abelian variety, the general statement you are looking for is the following:

Proposition 3.7 in https://arxiv.org/abs/1909.12187

Let $$L/k$$ be an extension of algebraically closed fields of characteristic zero. Let X be a proper variety over $$k$$. Then $$(\Delta_X^{gr})_L = \Delta^{gr}_{X_L}$$.

• I'm thinking about the following example: if $X=E\times C$ where $E$ is an elliptic curve and $C$ is a curve of genus $2$. Then $Sp(X)$ is supposed to be $X$. But the union of translates by closed points seems not including the codim $2$ generic point? I'm probably missing a point here. Could you help me out? Apr 1, 2020 at 15:48
• @ggttttll Sp(X) is scheme-theoretically really the closure of the closed subset (of the variety $X(k)$) given by the union of translates of positive-dimensional abelian subvarieties. It is then clear that $Sp(X) = X$ as a scheme if $X = E \times C$. Apr 1, 2020 at 16:12
• Thanks for all the help! Take care! Apr 1, 2020 at 16:54