The points of a cubic irreducible hypersurface of \({\mathbb{P}}^ n_ k\) as complete intersection subset
Let F be an irreducible reduced cubic hypersurface of \({\mathbb{P}}^ n_ k\), \(n\geq 3\), defined over an algebraically closed field k of char k\(=2\) and assume that F is not a cone whose vertex has dimension n-3. The main result states that through any point \(p\in F\) there passes a plane curve \(C\subset {\mathbb{P}}^ n\) with deg \(C\leq 4\) such that \(F\cdot C=(3\cdot \deg C)p.\) Since C is a complete intersection this means that every point of F is set-theoretically complete intersection (s.t.c.i.), being defined in the homogeneous coordinate ring of F by an ideal with n-1 generators. This fact together with a result of \textit{E. Stagnaro} [Atti Accad. Ligure Sci. Lett. 31 (1974), 138-148 (1975; Zbl 0363.14014)] implies the projective classification of cubic surfaces of \({\mathbb{P}}^ 3_ k\) on which any algebraic subvariety is s.t.c.i. The author also shows that the main result of the paper fails for a cubic cone whose vertex has dimension n-3 in case the ground field k is uncountable with char k\(=0\).
projective classification of cubic surfaces
scientific article; zbMATH DE number 3853267
scientific article