On efficient generation of pull-back of {$T\sb {\Bbb P\sp n}(-1)$}

Abstract

Let $f: X\to \mathbb P^n$ be a proper map such that dimension of $f(X)\ge 2$. We address the following question: Is $\dim H^o(X,\,f^{\ast}(T_{\mathbb P^n}(-1)) = n + 1$? We provide an affirmative answer under standard mild restrictions on $X$. We also point out that this provides an affirmative answer to a similar question raised via regular alteration of a closed subvariety in a blow-up of a regular local ring at its closed point in the mixed characteristics.