an:03865483
Zbl 0544.14028
Knop, F.; Lange, H.
Commutative algebraic groups and intersections of quadrics
EN
Math. Ann. 267, 555-571 (1984).
00148263
1984
j
14L10 14E25
completion of algebraic group; projectively normal embedding; intersection of quadrics
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.