RMC - Vol. 17 Num. 2, Johannes Huismann

Full paper in PDF:
$% J. Huisman, Real cubic hypersurfaces and group laws, Rev. Mat. Complut. 17 (2004), 395–401.%$

Real Cubic Hypersurfaces and Group Laws

Johannes HUISMAN

Département de Mathématiques
UFR Sciences et Techniques
Université de Bretagne Occidentale
6, avenue Victor Le Gorgeu
B.P. 809
29285 Brest Cedex — France

johannes.huisman@univ-brest.fr

Received: January 20, 2003
Accepted: Febrero 19, 2004

ABSTRACT

Let $X$ be a real cubic hypersurface in $Pn$ . Let $C$ be the pseudo-hyperplane of $X$ , i.e., $C$ is the irreducible global real analytic branch of the real analytic variety $X(R)$ such that the homology class $[C]$ is nonzero in $Hn -1(Pn(R),Z/2Z)$ . Let $L$ be the set of real linear subspaces $L$ of $Pn$ of dimension $n- 2$ contained in $X$ such that $L(R) (_ C$ . We show that, under certain conditions on $X$ , there is a group law on the set $L$ . It is determined by $' '' L +L +L = 0$ in $L$ if and only if there is a real hyperplane $H$ in $n P$ such that $' '' H .X = L+ L + L$ . We also study the case when these conditions on $X$ are not satisfied.

Key words: real cubic hypersurface, real cubic curve, real cubic surface, pseudo-hyperplane, pseudo-line, pseudo-plane, linear subspace, group.
2000 Mathematics Subject Classification: 14J70, 14P25.