Sets of points and their conductor
Let \(X\) be a set of distinct \(k\)-rational points in the \(n\)-dimensional projective space. If \(P\in X\) the minimal degree of an hypersurface which vanishes at every points of \(X\setminus\{P\},\) but not at \(P\) is called the separating degree of \(P.\) The socle values of \(X,\) \(\xi_1>\xi_2>\ldots>\xi_t,\) are the possible values of the separating degrees of the points of \(X.\) For every \(j\) let \(X_j\) be the subset of \(X\) whose points have separating degree equal to \(\xi_j.\) In this paper the author determines the ideal \(I(X_1\cup\ldots\cup X_j)\) for every \(j.\) Moreover, when \(X\subset{\mathbb P}^2,\) the author finds a sharp lower bound for the dimension of the algebra defined by the conductor of the coordinate ring of \(X.\)
scientific article; zbMATH DE number 2136610
scientific article