Rees algebras on smooth schemes: integral closure and higher differential operator

Villamayor, U Orlando

doi:10.4171/rmi/534

Cited by 23 publications

(37 citation statements)

References 10 publications

(6 reference statements)

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“…This definition coincides with the one given in Definition 3.6 (see [33,Proposition 4.4]). In fact if Diff(G) is the differential algebra generated by a Rees algebra G then…”

Section: Differential Algebras and Singular Locusmentioning

confidence: 71%

“…(see [33,Definition 4.2]). This definition coincides with the one given in Definition 3.6 (see [33,Proposition 4.4]).…”

Section: Differential Algebras and Singular Locusmentioning

confidence: 99%

“…Let G be a Rees algebra on a smooth scheme V over a field k. There is a natural way to construct a differential algebra containing G with the property of being the smallest differential algebra containing it (see [33,Theorem 3.4]). This Rees algebra will be denoted by Diff(G).…”

Section: The Differential Algebra Generated By a Rees Algebramentioning

confidence: 99%

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Singularities in positive characteristic, stratification and simplification of the singular locus

Bravo

Villamayor

2010

Advances in Mathematics

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“…This definition coincides with the one given in Definition 3.6 (see [33,Proposition 4.4]). In fact if Diff(G) is the differential algebra generated by a Rees algebra G then…”

Section: Differential Algebras and Singular Locusmentioning

confidence: 71%

“…(see [33,Definition 4.2]). This definition coincides with the one given in Definition 3.6 (see [33,Proposition 4.4]).…”

Section: Differential Algebras and Singular Locusmentioning

confidence: 99%

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Singularities in positive characteristic, stratification and simplification of the singular locus

Bravo

Villamayor

2010

Advances in Mathematics

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“…included in every other Diff-algebra containing it). We refer here to Theorem 3.4 in [39] for the proof, which follows easily from the argument for the one-variable case (Theorem 2.7). Let us indicate that if we fix a closed point x ∈ Z, and a regular system of parameters {x 1 , .…”

Section: Theorem 34 Assume Thatmentioning

confidence: 97%

“…These connections, and various other aspects, are studied in detail within Kawanoue's recent paper [26]. See also [38,39].…”

Section: Let (S M) Denote a Local Ring And Fix A Monic Polynomial Fmentioning

confidence: 98%

Hypersurface singularities in positive characteristic

Villamayor

2007

Advances in Mathematics

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We present results on multiplicity theory. Differential operators on smooth schemes play a central role in the study of the multiplicity of an embedded hypersurface at a point. This follows from the fact that the multiplicity is defined by the Taylor development of the defining equation at such point; and the Taylor development involves higher order differentials. On the other hand, the multiplicity of a hypersurface can be expressed in terms of general projections defined at étale neighborhoods of such point: Given a local embedding of a (d − 1)-dimensional hypersurface in a smooth d-dimensional scheme, a general projection on a smooth (d − 1)-dimensional scheme is a finite map on the hypersurface; and the multiplicity at the point is the degree of this finite map.We relate both approaches. In fact we study invariants of embedded hypersurfaces, defined in terms of differential operators, which express properties of the ramification of the morphism. Of particular interest is the case of hypersurfaces over fields of positive characteristic.A central result in multiplicity theory of hypersurfaces is a form of elimination of one variable in the description of highest multiplicity locus. This form of elimination, known over fields of characteristic zero, is achieved with the notion of Tschirnhausen polynomial introduced by Abhyankar. This is a key point in the proof of embedded desingularization.In this paper we provide a characteristic free approach to this form of elimination. Our alternative approach is based on projections on smooth (d − 1)-dimensional schemes.The properties of this new form of elimination remain weaker in positive characteristic, than it does in characteristic zero, when it comes to compatibility of elimination with permissible monoidal transformation. This opens the way to new questions on resolution problems. We also discuss the behavior of invariants, attached to a singularity at a point, with this form of elimination.

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Desingularization of binomial varieties in arbitrary characteristic. Part I. A new resolution function and their properties

Blanco

2012

Mathematische Nachrichten

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In this paper we construct a resolution function that will provide an algorithm of resolution of singularities for binomial ideals, over a field of arbitrary characteristic. For us, a binomial ideal means an ideal generated by binomial equations without any restriction, including monomials and p-th powers, where p is the characteristic of the base field.This resolution function is based in a modified order function, called E-order. The E-order of a binomial ideal is the order of the ideal along a normal crossing divisor E. The resolution function allows us to construct an algorithm of E-resolution of binomial basic objects, that will be a subroutine of the main resolution algorithm.

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Rees algebras on smooth schemes: integral closure and higher differential operator

Cited by 23 publications

References 10 publications

Singularities in positive characteristic, stratification and simplification of the singular locus

Singularities in positive characteristic, stratification and simplification of the singular locus

Hypersurface singularities in positive characteristic

Desingularization of binomial varieties in arbitrary characteristic. Part I. A new resolution function and their properties

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