Dicritical divisors after S.S. Abhyankar et I. Luengo

In [AL11], S.S Abhyankar and I. Luengo introduce a new theory of dicritical divisors in the most general framework. Here we simplify and generalize their results (see Theorems 3.1 and 3.2)

Resolution of Singularities of Arithmetical Threefolds II

We prove Grothendieck's Conjecture on Resolution of Singulari-ties for quasi-excellent schemes X of dimension three and of arbitrary characteristic. This applies in particular to X = SpecA, A a reduced complete Noetherian local ring of dimension three and to algebraic or arithmetical varieties of dimension three. Similarly, if F is a number field, a complete discretely valued field or more generally the quotient field of any excellent Dedekind domain O, any regular projective sur-face X/F has a proper and flat model X over O which is everywhere regular.Comment: updates and extends 'Resolution of Singularities of Arithmetical Threefolds I' posted on this websit

Constancy of the Hilbert-Samuel function

The Hilbert-Samuel function and the multiplicity function are fundamentallocally defined invariants on Noetherian schemes. They havebeen playing an important role in desingularization for many years.Bennett studied upper semicontinuity of the Hilbert-Samuel functionon schemes and proved that it is non increasing under permissibleblowing ups. The latter are blowing ups at regular subschemes alongwhich the singular scheme is normally flat.For a reduced scheme, the Hilbert-Samuel function is constant ifand only if it is regular: this translates the question of resolutionof singularities into a problem of lowering the Hilbert-Samuel function.We show here that, this result can be extended to non reducedschemes, as follows: Given a locally Noetherian scheme X such thatthe local rings are excellent for every point, then the Hilbert-Samuelfunction is constant on X if and only if X is normally flat along itsreduction and the reduction itself is regular

Existence des diviseurs dicritiques, d'apr\`es S.S.Abhyankar

In geometric terms, given a singular foliation of the plane, a dicritical divisor is (whenever it exists) an irreducible component of the exceptional divisor which is transverse to the foliation. Abhyankar gave recently a definition of the dicritical divisors which generalize and algebraicize the geometrical definition in the local case and the polynomial case. Following his work, we give a geometrical interpretation of these dicritical divisors and new proofs of their existence.Comment: 9 pages, in Frenc