# What does “Biextension of Abelian Varieties” mean?

If I have two schemes $X$ and $Y$, which are such that my question makes sense (I guess, they should be abelian varieties over a field $k$, so assume this).

Then I have often read, but nowhere found a proper definition, the notion of a $\mathbb{G}_{m}$- biextension of $X \times Y$.

I would be very glad if someone could explain this notion properly to me, maybe with an example (I think one often speaks of the "Poincaré-Biextension).

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## 1 Answer

The definition is not often written down as it is rather messy. A precise definition of a biextension can be found in SGA7.1, exposees VII and VIII. The Poincare Biextension is discussed in the article of Moret-Bailly in Asterisque 127 (though this reference is in no way canonical).

I am sorry that this answer is very minimal, but given how long the question has been sitting here I thought something was better than nothing!

David Holmes

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Welcome, David! Why don't you register on this site properly? It would be good to have you around. –  Alex B. Jan 7 '13 at 11:41