Ridge of the tangent cone of a singularity: a forgotten theorem
The authors give a proof of a theorem that, according to them, Jean Giraud ``forgot'' to state. The situation is the following one: let \(X\) be an excellent scheme and let \(H_X\) denote the modified Hilbert-Samuel function of \(X\). It is known that this function cannot increase under permissible blowing ups and that it is upper semi-continuous. The result proved in this paper is a strengthened version of this statement. More precisely, let us assume that \(X\) is embedded in some regular scheme. Then for every \(x\in X\) let us denote by \(\tau_{st}(x)\) the codimension of the ridge of the tangent cone of \(X\) at \(x\) (the ridge of the tangent cone is the largest additive group that preserves the tangent cone by translation). Then the function \[ \iota : X\longrightarrow \mathbb N^{\mathbb N}\times -\mathbb N\qquad \] \[ \qquad x\longmapsto (H_X(x),-\tau_{st}(x)) \] does not increase under permissible blowing ups for the lexicographic ordering and it is upper semi-continuous. When \(X\) is not assumed to be embedded in some regular scheme, the authors proves an analogue of this statement by replacing \(\tau_{st}\) by a modified version of this codimension.
scientific article; zbMATH DE number 6708166