On the classification of non-normal cubic hypersurfaces

Abstract

In this article we study the classification of non-normal cubic hypersurfaces over an algebraically closed field K of arbitrary characteristic. Let X ⊂ P K r be an irreducible non-normal cubic hypersurface. If r ≥ 5 , then X is necessarily a cone (Remark 2.3). In view of this fact it suffices to classify irreducible non-normal cubic hypersurfaces X ⊂ P K r for r ≤ 4 . We prove that there are precisely five non-normal cubic equations (resp. six non-normal cubic equations) when char K ≠ 2 , 3 (resp. when char K is either 2 or 3), up to projective equivalence. Also we describe the normalization of X in detail.