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Additive actions on projective hypersurfaces. (English) Zbl 1326.14112
Cheltsov, Ivan (ed.) et al., Automorphisms in birational and affine geometry. Papers based on the presentations at the conference, Levico Terme, Italy, October 29 – November 3, 2012. Cham: Springer (ISBN 978-3-319-05680-7/hbk; 978-3-319-05681-4/ebook). Springer Proceedings in Mathematics & Statistics 79, 17-33 (2014).
Summary: By an additive action on a hypersurface \(H\) in \(\mathbb P^{n+1}\) we mean an effective action of a commutative unipotent group on \(\mathbb P^{n+1}\) which leaves \(H\) invariant and acts on \(H\) with an open orbit. B. Hassett and Y. Tschinkel [Int. Math. Res. Not. 1999, No. 22, 1211–1230 (1999; Zbl 0966.14033)] have shown that actions of commutative unipotent groups on projective spaces can be described in terms of local algebras with some additional data. We prove that additive actions on projective hypersurfaces correspond to invariant multilinear symmetric forms on local algebras. It allows us to obtain explicit classification results for non-degenerate quadrics and quadrics of corank one.
For the entire collection see [Zbl 1291.14005].

MSC:
14L30 Group actions on varieties or schemes (quotients)
13H99 Local rings and semilocal rings
15A69 Multilinear algebra, tensor calculus
14J50 Automorphisms of surfaces and higher-dimensional varieties
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