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On efficient generation of pull-back of $$T_{\mathbb P^n}(-1)$$. (English) Zbl 1127.14016
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 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 $$[m]$$.
##### MSC:
 14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials 14B05 Singularities in algebraic geometry 13H15 Multiplicity theory and related topics
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