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

Some remarks about additive polynomials of degree p in characteristic p>0

International audienceAdditive polynomials play an important role in the local study of singularities in positive characteristic. For a base field k, the Hilbert-Samuel stratum of an affine cone over k is a linear space if k is perfect, but this is no longer true in general over non-perfect fields. H. Hironaka introduced additive subgroups in 1970 in order to quantify this phenomenon but little is known about classifying them except in small dimension.Building on joint works with A. Benito and A. Reguera on arc spaces, and with V. Cossart and B. Schober on the Hilbert-Samuel stratum, I will point out some of the questions arising for hypersurface cones defined by an additive polynomial of degree p in characteristic p>0

An axiomatic version of Zariski's patching theorem

30 pagesWe state six axioms concerning any regularity property P in a given birational equivalence class of algebraic threefolds. Axiom 5 states the existence of a local uniformization in the sense of valuations for P. If axioms 1 to 4 are satisfied by P, then the function field has a projective model which is everywhere regular w.r.t. P. Axiom 6 ensures the existence of P-resolution of singularities for any projective model. Applications concern resolution of singularities of vector fields and a weak version of Hironaka's Strong Factorization Conjecture for birational morphisms of nonsingular projective threefolds, both of them in characteristic zero

Resolution of singularities of threefolds in mixed characteristic: case of small multiplicity

pages 1-39; The final publication is available at www.springerlink.com DOI: 10.1007/s13398-012-0103-5International audienceThis article is part of the authors' program on Resolution of Singularities of threefolds in mixed characteristic. We prove Local Uniformization for rank one valuations centered in a reduced hypersurface singularity $(X,x)\subset (Z,x)$, $Z$ any regular excellent fourfold, under the following assumption: $X$ has multiplicity smaller than the residue characteristic of $Z$. This is obtained by blowing up regular centers which are locally Hironaka permissible at the center of the valuation