## Commutative algebraic groups and intersections of quadrics.(English)Zbl 0544.14028

Let E be a commutative, connected algebraic group over an algebraically closed field k and $$0\to G\to E\to^{\pi}A\to 0$$ the canonical exact sequence with a connected linear group G and an abelian variety A. (The assumption on k should be added to the paper.) Each equivariant completion $$\bar G$$ of G induces a completion $$\bar E:=E\times^ G\bar G$$ of E and a G-linearized linebundle L on $$\bar G$$ induces a linebundle E(L) on $$\bar E$$. The purpose of the paper is to give criteria when a linebundle of the form $$E(L)\otimes \pi^*(N)\quad(N\in Pic A)$$ defines a projectively normal embedding of $$\bar E$$ into projective space and whether the image is an intersection of quadrics. Then the results are specialized to the case when $$\bar G$$ is a product of projective spaces. In this section all equivariant completions $$\bar G$$ are determined where $$\bar G$$ is isomorphic to a projective space.

### MSC:

 14L10 Group varieties 14E25 Embeddings in algebraic geometry
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### References:

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