Resolution of Singularities of an Algebraic Variety Over a Field of Characteristic Zero: II

1964 ◽  
Vol 79 (2) ◽  
pp. 205 ◽  
Author(s):  
Heisuke Hironaka
Keyword(s):  
Algebraic Variety ◽  
Characteristic Zero ◽  
Resolution Of Singularities

A Module-theoretic Characterization of Algebraic Hypersurfaces

2018 ◽  
Vol 61 (1) ◽  
pp. 166-173
Author(s):  
Cleto B. Miranda-Neto
Keyword(s):  
Algebraic Variety ◽  
Characteristic Zero ◽  
Vector Fields ◽  
Exterior Power ◽  
Algebraic Hypersurfaces ◽  
Theoretic Characterization

AbstractIn this note we prove the following surprising characterization: if X ⊂ is an (embedded, non-empty, proper) algebraic variety deûned over a field k of characteristic zero, then X is a hypersurface if and only if the module of logarithmic vector fields of X is a reflexive -module. As a consequence of this result, we derive that if is a free -module, which is shown to be equivalent to the freeness of the t-th exterior power of for some (in fact, any) t ≤ n, then necessarily X is a Saito free divisor.

A STRENGTHENING OF RESOLUTION OF SINGULARITIES IN CHARACTERISTIC ZERO

2003 ◽  
Vol 86 (2) ◽  
pp. 327-357 ◽  
Author(s):  
A. BRAVO ◽  
O. VILLAMAYOR U.
Keyword(s):  
Characteristic Zero ◽  
Resolution Of Singularities ◽  
Closed Subscheme ◽  
Subject Classification ◽  
Invertible Sheaf ◽  
Simple Manner ◽  
Birational Morphism ◽  
Exceptional Locus

Let $X$ be a closed subscheme embedded in a scheme $W$, smooth over a field ${\bf k}$ of characteristic zero, and let ${\mathcal I} (X)$ be the sheaf of ideals defining $X$. Assume that the set of regular points of $X$ is dense in $X$. We prove that there exists a proper, birational morphism, $\pi : W_r \longrightarrow W$, obtained as a composition of monoidal transformations, so that if $X_r \subset W_r$ denotes the strict transform of $X \subset W$ then:(1) the morphism $\pi : W_r \longrightarrow W$ is an embedded desingularization of $X$ (as in Hironaka's Theorem);(2) the total transform of ${\mathcal I} (X)$ in ${\mathcal O}_{W_r}$ factors as a product of an invertible sheaf of ideals ${\mathcal L}$ supported on the exceptional locus, and the sheaf of ideals defining the strict transform of $X$ (that is, ${\mathcal I}(X){\mathcal O}_{W_r} = {\mathcal L} \cdot {\mathcal I}(X_r)$).Thus (2) asserts that we can obtain, in a simple manner, the equations defining the desingularization of $X$.2000 Mathematical Subject Classification: 14E15.

scholarly journals Higher Nash blowups

2007 ◽  
Vol 143 (6) ◽  
pp. 1493-1510 ◽  
Author(s):  
Takehiko Yasuda
Keyword(s):  
Algebraic Variety ◽  
Characteristic Zero ◽  
Negative Integer ◽  
Curve Singularity ◽  
Irreducible Curve ◽  
Numerical Monoid

AbstractFor each non-negative integer n we define the nth Nash blowup of an algebraic variety, and call them all higher Nash blowups. When n=1, it coincides with the classical Nash blowup. We study higher Nash blowups of curves in detail and prove that any curve in characteristic zero can be desingularized by its nth Nash blowup with n large enough. Moreover, we completely determine for which n the nth Nash blowup of an analytically irreducible curve singularity in characteristic zero is normal, in terms of the associated numerical monoid.

scholarly journals Differential operators on a hypersurface

1986 ◽  
Vol 103 ◽  
pp. 67-84 ◽  
Author(s):  
Balwant Singh
Keyword(s):  
Algebraic Variety ◽  
Characteristic Zero ◽  
Differential Operators ◽  
Coordinate Ring ◽  
Irreducible Curve ◽  
Additional Hypothesis

We study differential operators on an affine algebraic variety, especially a hypersurface, in the context of Nakai’s Conjecture. We work over a field k of characteristic zero. Let X be a reduced affine algebraic variety over k and let A be its coordinate ring. Let be the A-module of differential operators of A over k of order ≤ n. Nakai’s Conjecture asserts that if is generated by for every n ≥ 2 then A is regular. In 1973 Mount and Villamayor [6] proved this in the case when X is an irreducible curve. In the general case no progress seems to have been made on the conjecture, except for a result of Brown [2], where the assertion is proved under an additional hypothesis. An interesting result proved by Becker [1] and Rego [8] says that Nakai’s Conjecture implies the Conjecture of Zariski-Lipman, which is still open in the general case and which asserts that if the module of k-derivations of A is A-projective then A is regular.

scholarly journals Corrections to “Seminormal rings and weakly normal varieties”

1987 ◽  
Vol 107 ◽  
pp. 147-157 ◽  
Author(s):  
Marie A. Vitulli
Keyword(s):  
Algebraic Variety ◽  
Characteristic Zero ◽  
Regular Function ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Regular Functions ◽  
Normal Varieties

In “Seminormal rings and weakly normal varieties” we introduced the notion of a c-regular function on an algebraic variety defined over an algebraically closed field of characteristic zero. Our intention was to describe those k-valued functions on a variety X that become regular functions when lifted to the normalization of X, but without any reference to the normalization in the definition. That is, we aspired to identify the c-regular functions on X with the regular functions on the weak normalization of X

scholarly journals Tensor Structure on Smooth Motives

2011 ◽  
Vol 9 (1) ◽  
pp. 57-101 ◽  
Author(s):  
Anandam Banerjee
Keyword(s):  
Characteristic Zero ◽  
Full Subcategory ◽  
Resolution Of Singularities ◽  
Homotopy Category ◽  
Tensor Structure ◽  
Perfect Field ◽  
Base Scheme

AbstractRecently, Bondarko constructed a DG category of motives, whose homotopy category is equivalent to Voevodsky's category of effective geometric motives, assuming resolution of singularities. Soon after, Levine extended this idea to construct a DG category whose homotopy category is equivalent to the full subcategory of motives over a base-scheme S generated by the motives of smooth projective S-schemes, assuming that S is itself smooth over a perfect field. In both constructions, the tensor structure requires ℚ-coefficients. In this article, it is shown how to provide a tensor structure on the homotopy category mentioned above, when S is semi-local and essentially smooth over a field of characteristic zero. This is done by defining a pseudo-tensor structure on the DG category of motives constructed by Levine, which induces a tensor structure on its homotopy category.

Exceptional divisors that are not uniruled belong to the image of the Nash map

2011 ◽  
Vol 11 (2) ◽  
pp. 273-287 ◽  
Author(s):  
Monique Lejeune-Jalabert ◽  
Ana J. Reguera
Keyword(s):  
Irreducible Component ◽  
Characteristic Zero ◽  
Exceptional Divisor ◽  
Resolution Of Singularities ◽  
Closed Field ◽  
Algebraically Closed Field

AbstractWe prove that, ifXis a variety over an uncountable algebraically closed fieldkof characteristic zero, then any irreducible exceptional divisorEon a resolution of singularities ofXwhich is not uniruled, belongs to the image of the Nash map, i.e. corresponds to an irreducible component of the space of arcs$X_\infty^{\mathrm{Sing}}$onXcentred in SingX. This reduces the Nash problem of arcs to understanding which uniruled essential divisors are in the image of the Nash map, more generally, how to determine the uniruled essential divisors from the space of arcs.

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