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 Pn such that H · X = L + L' + L''. We also study the case when these conditions on X are not satisfied.
@article{urn:eudml:doc:44524,
title = {Real cubic hypersurfaces and group laws.},
journal = {Revista Matem\'atica de la Universidad Complutense de Madrid},
volume = {17},
year = {2004},
pages = {395-401},
zbl = {1067.14039},
mrnumber = {MR2083961},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:44524}
}
Huisman, Johannes. Real cubic hypersurfaces and group laws.. Revista Matemática de la Universidad Complutense de Madrid, Tome 17 (2004) pp. 395-401. http://gdmltest.u-ga.fr/item/urn:eudml:doc:44524/