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Vincent Cossart

Publications and source records attributed to Vincent Cossart.

9 recordsLinked to original sources
arXiv2023

Constancy of the Hilbert-Samuel function

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

math.AG↗
arXiv2020

Characteristic polyhedra of singularities without completion -- Part II

Hironaka's characteristic polyhedron is an important combinatorial object reflecting the local nature of a singularity. We prove that it can be determined without passing to the completion if the local ring is a G-ring and if additionally either it is Henselian, or a certain polynomiality condition $ (\mathrm{Pol}) $ holds, or a mild condition $(*) $ on the singularity holds. For example, the latter is fulfilled if the residue field is perfect.

math.AG↗
arXiv2019

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.

math.AG↗
arXiv2018

Invariance of Hironaka's characteristic polyhedron

We show that given a face of Hironaka's characteristic polyhedron, it does only depend on the singularity and a flag defined by the linear form determining the face. As a consequence we get that certain numerical data obtained from the characteristic polyhedron are invariants of the singularity. In particular, they do not depend on an embedding.

math.AG↗
arXiv2014

Characteristic polyhedra of singularities without completion

Let $(R,M,k)$ be a regular local G-ring with regular system of parameters $(u_1, \ldots ,u_d,y)$. We prove that the Hironaka characteristic polyhedron $Δ(f;u_1, \ldots ,u_d)$, $f \not \in (u_1, \ldots ,u_d)$ of a hypersurface singularity $X={\rm Spec}R/(f)$ can be computed in some system of coordinates belonging to $R$. No assumption on the residue characteristic is required.

math.AG↗
arXiv2013

Existence des diviseurs dicritiques, d'après 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.

math.AG↗
arXiv2013

Canonical embedded and non-embedded resolution of singularities for excellent two-dimensional schemes

We prove the existence of resolution of singularities for arbitrary (not necessarily reduced or irreducible) excellent two-dimensional schemes, via permissible blow-ups. The resolution is canonical, and functorial with respect to automorphisms or etale or Zariski localizations. We treat the embedded case as well as the non-embedded case, with or without a boundary, and we relate the diferent versions. In the non-embedded case, a boundary is a collection of locally principal closed subschemes. Our main tools are the stratifications by Hilbert-Samuel functions and the characteristic polyhedra introduced by H. Hironaka. In an appendix we show that the standard method used in characteristic zero - the theory of maximal contact - does not work for surfaces in positive characteristic (the counterexamples are hypersurfaces in affine threespace and work over any field of positive characteristic). In this new version, we treat the case of locally noetherian but not necessarily noetherian schemes in an appropriate way. Here one does not have a finite resolution sequence, but still a canonical resolution morphism by glueing. The same techniques allow to treat algebraic spaces and stacks.

math.AG↗