Real Cubic Hypersurfaces and Group Laws - Huisman, Johannes (Revista Matemática Complutense, 2004)
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 + L0 + L00 = 0 in L if and only if there is a real hyperplane H in Pn such that H • X = L + L0 + L00. We also study the case when these conditions on X are not satisfied.