Timeline for Given a semi-abelian scheme, is the set of points such that the fibres are abelian varities open?
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 23, 2019 at 21:23 | vote | accept | Z Wu | ||
| Mar 23, 2019 at 14:47 | comment | added | Jason Starr | @LanceWu. Surjectivity of a morphism of schemes can be checked on geometric points. | |
| Mar 23, 2019 at 14:43 | comment | added | Z Wu | In lemma 1, can you eleborate more on the part $h\in U_L^{-1}$, I think it is a bit handwaving there, I know they are suppose to be $L$-rational points considering the morphism $G_L\times_L G_L\rightarrow G_L$ induces the mulitplication map $G_L(L)\times G_L(L)\rightarrow G_L(L)$. Can we show that if $g\cdot {U_L}^{-1}\cap U_L$ is non-empty then it contains a $L$-rational point? I guess this is where algebraic closure comes in. Also the lemma seems only prove the surjectivity of geometrical points. | |
| Mar 23, 2019 at 14:23 | history | edited | Jason Starr | CC BY-SA 4.0 |
added 10 characters in body
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| Mar 23, 2019 at 14:00 | comment | added | Jason Starr | @LanceWu. Quasi-compactness of a morphism can be checked after replacing the target by each open in an open affine covering of the target (and replacing the domain by the inverse image open). Surjectivity of a morphism can be checked on geometric points. Every geometric point of $G$ maps to a geometric point of $S$. Thus, it suffices to check surjectivity after restricting to the fiber in $G_S$ over every geometric point of $S$. | |
| Mar 23, 2019 at 12:15 | comment | added | Z Wu | In proposition 3, it seem one didn't need to "prove this locally on the target" since you didn't use this condition. Also, why does the surjectivity on fibers over geometrical points of $V$ implies the surjectivity of $m_U$. | |
| S Mar 23, 2019 at 11:20 | history | answered | Jason Starr | CC BY-SA 4.0 | |
| S Mar 23, 2019 at 11:20 | history | made wiki | Post Made Community Wiki by Jason Starr |