Timeline for What is a "non-trivial" example of a commutative algebraic group over $\mathbb{C}$?
Current License: CC BY-SA 4.0
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| Nov 9, 2021 at 16:52 | comment | added | David E Speyer | I changed a $C$ to a $D$ at the end of the first paragraph, you might want to check that I got it right. | |
| Nov 9, 2021 at 16:52 | history | edited | David E Speyer | CC BY-SA 4.0 |
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| Nov 9, 2021 at 15:52 | comment | added | YCor | Yes indeed: since starting from $G_a^n\times A$, $A$ abelian variety, every finite normal subgroup is contained in $A$ and hence the quotient also splits. | |
| Nov 9, 2021 at 15:15 | vote | accept | Gabriel | ||
| Nov 9, 2021 at 15:05 | comment | added | Will Sawin | @YCor If the multiplicities are one then difference between any two points of $D$ Is torsion in the Jacobian then this torsor will split after passing to a finite cover. If this is not true, it won't split. A non-split extension by $\mathbb G_a$ never becomes split after passing to a finite cover, I don't think. | |
| Nov 9, 2021 at 15:01 | comment | added | YCor | Also, do your examples remain non-split on every finite cover? (I'm mentioning, as a comment to the question, a non-split example which by construction becomes split after passing to a finite connected cover.) | |
| Nov 9, 2021 at 14:59 | comment | added | Will Sawin | @Ycor The argument I gave shows that extension by $\mathbb G_m$ produced by generalized Jacobians when the divisor $D$ contains two different points are nontrivial, but I believe taking the derivative of this argument shows that extensions by $\mathbb G_a$ produced by generalized Jacobians when $D$ contains a point of multiplicity $\geq 2$ are nontrivial. | |
| Nov 9, 2021 at 14:55 | comment | added | YCor | Can you get both non-split extensions with kernel $G_a$ and $G_m$? | |
| Nov 9, 2021 at 14:53 | comment | added | Will Sawin | @Gabriel I guess the simplest example would be the generalized Jacobian associated to two points on a curve of genus $1$, which is an extension of that elliptic curve by $\mathbb G_m$. As a scheme, it is some nontrivial degree-0 line bundle on the elliptic curve with the zero-section removed, but it carries a group structure. | |
| Nov 9, 2021 at 14:38 | comment | added | Gabriel | Dear @WillSawin, that's incredible. Since I never studied generalized Jacobians, do you have a simple example of it in mind? | |
| Nov 9, 2021 at 14:30 | history | answered | Will Sawin | CC BY-SA 4.0 |