Open AccessReal Cubic Hypersurfaces and Group Laws
Author(s) -
Johannes Huisman
Publication year - 2004
Publication title -
revista matemática complutense
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.093
H-Index - 24
eISSN - 1696-8220
pISSN - 1139-1138
DOI - 10.5209/rev_rema.2004.v17.n2.16739
Subject(s) - hyperplane , hypersurface , linear subspace , mathematics , combinatorics , cubic function , group (periodic table) , dimension (graph theory) , mathematical analysis , pure mathematics , physics , quantum mechanics
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.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