On efficient generation of pull-back of \(T_{\mathbb P^n}(-1)\) (Q2382934)

Let \(K\) be an algebraically closed field and \(X\) a reduced connected scheme of finite type over \(K\). In the paper under review, it is shown that for a proper map \(f:\;X\rightarrow {\mathbb P}^n\), the space \(H^0(X,f^*(T_{{\mathbb P}^n}(-1)))\) is \(n+1\)-dimensional, under the assumption that the image under \(f\) of each irreducible component of \(X\) is at least \(2\)-dimensional, and that the image of the intersection of two components is either empty or at least \(1\)-dimensional. By work of \textit{G. Hein} [Rocky Mt. J. Math. 30, No. 1, 217--235 (2000; Zbl 0983.14011)] it is known that this cannot be generalized to maps with \(1\)-dimensional image. Via Zariski's main theorem, this result is used to address a problem motivated by the study of intersection multiplicities over the blow-up of a regular local ring at its closed point in the mixed characteristics. Specifically, it is shown that for an \(n\)-dimensional regular local ring \((R,m)\) with residue field \(K\), then \(H^0(Y,T_{{\mathbb P}^{n-1}}(-1)\otimes {\mathcal O}_Y)\) is \(n\)-dimensional if \(Y\) is the reduced scheme associated to a regular alteration of the special fiber of the blow-up of a subvariety of \(\text{Spec} R\) at the point \([m]\).