Real Cubic Hypersurfaces and Group Laws

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 Pn such that H · X = L + L + L . We also study the case when these conditions on X are not satisfied.