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Some remarks on compactifications of commutative algebraic groups. (English) Zbl 0587.14030
Let E be a connected commutative algebraic group defined on the algebraically closed field k. The authors wish to embed projectively E by first completing it and then embedding the compactification $$\bar E.$$ Let G be the maximal linear subgroup of E and $$A=E/G$$ the abelian variety quotient. If P is a projective G-variety and $$G\to P$$ an open G- immersion, then the bundle E(P) with fibre P over A associated to the principal G-bundle $$E\to A$$ is a compactification. - In fact, if X is an E-variety, $$i: E\to X$$ an open E-immersion, and $$\bar G$$ the closure of the G-orbit of i(0), there is an E-morphism $$E(\bar G)\to X$$. The authors prove that it is birational and proper, and an isomorphism if X is normal and $$\bar G$$ has only finitely many G-orbits. This shows that, under the latter conditions, the above compactifications are the only ones.
The G-linear projective embeddings of E(P) are related to the G-linear line bundles on E(P). For P complete, it is shown that the group of G- linear line bundles on E(P) is the product of the groups of G-linear line bundles on P and all line bundles on A. If k is the field of the complex numbers then $$G={\mathbb{G}}_ a^{(p)}\times {\mathbb{G}}_ m^{(q)}$$. A completion of G can be constructed by embedding each factor in $$P^ 1$$. Different factorizations of G can lead to different compactifications of E, as an example presented shows. Finally, if the compactification P of G comes from such a factorization, and the projective embedding of E uses a normally generated line bundle on A, the homogeneous ideal of the embedded E(P) is shown to be generated by forms of degree $$\leq \dim A+3.$$
Reviewer: A.R.Magid

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