Lifting the \(k\)-Buchsbaum property
Let \(X\subset\mathbb{P}^n\) be an equidimensional projective subscheme and \(F\) be a sufficiently general hypersurface of \(\mathbb{P}^n\). \(X\) is said to be \(k\)-Buchsbaum if \[ (x_0,x_1, \dots,x_n)^k \cdot\bigoplus_{t\in \mathbb{Z}}H^i\bigl( \mathbb{P}^n, {\mathcal I}_X(t) \bigr)=0 \] where \(K[x_0,x_1, \dots,x_n]\) is the coordinate ring of \(\mathbb{P}^n\) and \({\mathcal I}_X\) is the sheaf ideal of \(X\). In this paper the author considers relation between the \(k\)-Buchsbaum property for \(X\) and for \(F\cap X\). He also investigates consequences of \(k\)-Buchsbaum on the degree of \(F\).
scientific article; zbMATH DE number 1517306