Applications of the lifting from \(\text{char } p\) of some rational \(n\)-fold (Q1903687)

Let \(K\) be an algebraically closed field of characteristic \(p > 0\), \(\mathbb{P}^n\) the projective space over \(K\), \(A \subset \mathbb{P}^n\) a finite subset of \(s\) elements. Let \(X_{s,A}\) be the blowing-up of \(\mathbb{P}^n\) at \(A\). The purpose of this paper is to study Kodaira's vanishing theorem on \(X_{s,A}\). Following ideas of Fontaine and Messing in order to have Kodaira's vanishing theorem on \(X_{s,A}\) it is sufficient, when \(p > \dim X_{s,A}\), to lift \(X_{s,A}\) to the Witt vectors of the base field. The author proves this last statement and extends some results of \textit{A. V. Geramita}, \textit{A. Gimigliano} and \textit{Y. Pitteloud} [Math. Ann. 301, No. 2, 363-380 (1995; Zbl 0813.14008)] to characteristic \(p > 0\).