A note on adjoint linear systems

In dimension 2 there is a very useful criterion, due to Miles Reid, giving a necessary and sufficient condition for an adjoint linear system on a nonsingular complex projective surface X to be free at a point \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${p\in X}$$\end{document} , in terms of the coherent cohomology of certain divisors on the blowup of X at p. In this note we extend Miles Reid’s result to higher dimensions.