Closed subschemes of \(\mathbb{P}^ r\): Remarks on the cohomology of their 1- codimensional subschemes (Q1314265)

Let \(X \subset \mathbb{P}_ k^ r\) \((k\) an algebraically closed field) be a \(q+1\)-dimensional closed integral scheme whose canonical sheaf \(\omega_ X\) is invertible, and let \(Y\subset X\) be a 1-codimensional subscheme defined by the ideal \(J_ Y \subset {\mathcal O}_ X\). The first part of this paper is devoted to obtain an estimate for the least integer \(N\) such that \(H^{q + 1} (X,J_ Y(n) \otimes \omega_ X) = 0\) for all \(n > N\). For \(q = 0\), i.e. \(X\) a curve, the result is essentially that obtained by \textit{S. Greco} [Math. Ann. 284, No. 2, 343-351 (1989; Zbl 0645.14024)]. -- The case \(q>0\) is obtained from this by induction on \(q\). The second part of the paper uses the estimate to give cohomological criteria for \(Y\) to be the complete intersection of \(X\) with a hypersurface. The results obtained are related to and/or generalize those of \textit{B. Segre}, \textit{E. D. Davies} [Rend. Semin. Mat., Torino 43, 333- 353 (1985; Zbl 0675.14024) and in Curves Semin. Queen's, Vol. 3, Kingston 1993, Queen's Pap. Pure Appl. Math. 67, Exposé D (1984; Zbl 0596.14037)] and \textit{S. Greco} [loc. cit. and Semin. Geom., Univ. Stud. Bologna 1985, 89-100 (1986; Zbl 0613.14009)].