On the length of the absolute Samuel stratum (Q919424)

Let X be a scheme of finite type over \({\mathbb{C}}\). The following result is proved: Let x be a generic point of the center of a permissible monoidal transformation \(X'\to X\) and let y be any generic point of \(\Pr oj(T_ xX)\). If \(s_{X,x}\) is finite, then \(s_{X',y}<s_{X,x}\) or \(H^{(r)}_{X',y}<H_{X,x}\). - Here r is the transcendence degree of \({\mathbb{C}}(y)\) over \({\mathbb{C}}(x)\), \(s_{X,x}\) is the length of the Samuel stratum of X passing through x and \(H_{X,x}\) is the Hilbert-Samuel function. The technique of the proof has applications to the theory of maximal contact.