Showing papers in "Tohoku Mathematical Journal in 1981"

Desingularization of embedded excellent surfaces

Abstract: Introduction. In [5] the point at issue is reduced by Hironaka to the study of a singular point xeX isolated in its Samuel stratum. Then he makes the blowing-up X —»X with center x. The difficult case is when the directrix, i.e., the vector space of translations leaving stable the tangent cone to X at point x (as a scheme) is of dimension 2. Indeed, if the directrix is of dimension 1, the near points of the blowing-up with center x are rational, while if it is of dimension 0, there is none. We recall that a near point of X at x of a blowing-up X —» X is a point x e X' verifying the following equality between the Hilbert-Samuel series of X and X' at x and x': HX{X) = H.,(X')/0 T) , where d is the transcendence degree of the residue extension. In the case of the directrix being of dimension 2, Hironaka proves that an invariant denoted (β, ε, a) strictly decreases for the lexicographical ordering at any very near rational point. Here we show that at any very near algebraic non rational point the invariant β strictly decreases. We recall that a near algebraic point is very near if its directrix has the same dimension as that of x. We will not try to say more precisely here how the theorem below fits in with Hironaka's proof.