Regular and singular points of algebraic varieties

Form rings and regular sequences

Nagoya Mathematical Journal ◽

10.1017/s0027763000018225 ◽

1978 ◽

Vol 72 ◽

pp. 93-101 ◽

Cited By ~ 121

Author(s):

Paolo Valabrega ◽

Giuseppe Valla

Keyword(s):

Local Ring ◽

Singular Points ◽

Point Of View ◽

Algebraic Varieties ◽

Associated Graded Ring ◽

Graded Ring ◽

Standard Bases ◽

Regular Sequences ◽

Initial Forms ◽

The Ideal

Hironaka, in his paper [H1] on desingularization of algebraic varieties over a field of characteristic 0, to deal with singular points develops the algebraic apparatus of the associated graded ring, introducing standard bases of ideals, numerical characters ν* and τ* etc. Such a point of view involves a deep investigation of the ideal b* generated by the initial forms of the elements of an ideal A of a local ring, with respect to a certain ideal a.

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Number of Jordan blocks of the maximal size for local monodromies

Compositio Mathematica ◽

10.1112/s0010437x13007513 ◽

2014 ◽

Vol 150 (3) ◽

pp. 344-368

Author(s):

Alexandru Dimca ◽

Morihiko Saito

Keyword(s):

Spectral Sequence ◽

Hodge Structure ◽

Singular Points ◽

Mixed Hodge Structure ◽

Algebraic Varieties ◽

Isolated Singularity ◽

Singular Case ◽

Maximal Size ◽

Combinatorial Formulas ◽

Weight Spectral Sequence

AbstractWe prove formulas for the number of Jordan blocks of the maximal size for local monodromies of one-parameter degenerations of complex algebraic varieties where the bound of the size comes from the monodromy theorem. In the case when the general fibers are smooth and compact, the proof calculates some part of the weight spectral sequence of the limit mixed Hodge structure of Steenbrink. In the singular case, we can prove a similar formula for the monodromy on the cohomology with compact supports, but not on the usual cohomology. We also show that the number can really depend on the position of singular points in the embedded resolution, even in the isolated singularity case, and hence there are no simple combinatorial formulas using the embedded resolution in general.

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Regular and singular points of algebraic varieties

Introduction to Commutative Algebra and Algebraic Geometry ◽

10.1007/978-1-4612-5290-0_6 ◽

1985 ◽

pp. 163-195

Author(s):

Ernst Kunz

Keyword(s):

Singular Points ◽

Algebraic Varieties

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Binary diagrams of mesomorphous compounds exhibiting singular points

Annales de Physique ◽

10.1051/anphys/197803030381 ◽

1978 ◽

Vol 3 ◽

pp. 381-386 ◽

Cited By ~ 5

Author(s):

F. Hardouin ◽

G. Sigaud ◽

M.-F. Achard ◽

H. Gasparoux

Keyword(s):

Singular Points

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Effect of proximity of the Fermi level to singular points in the band structure on the kinetic and lattice properties of metals and alloys

Uspekhi Fizicheskih Nauk ◽

10.3367/ufnr.0154.198803g.0525 ◽

1988 ◽

Vol 154 (3) ◽

pp. 525 ◽

Cited By ~ 5

Author(s):

V.P. Antropov ◽

Valentin G. Vaks ◽

M.I. Katsnel'son ◽

V.G. Koreshkov ◽

A.I. Likhtenshtein ◽

...

Keyword(s):

Band Structure ◽

Fermi Level ◽

Metals And Alloys ◽

Singular Points

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Chow Rings, Decomposition of the Diagonal, and the Topology of Families (AM-187)

10.23943/princeton/9780691160504.001.0001 ◽

2017 ◽

Author(s):

Claire Voisin

Keyword(s):

Recent Work ◽

Chow Groups ◽

Ring Structure ◽

Complete Intersections ◽

Algebraic Varieties ◽

Chow Ring ◽

Hodge Structures ◽

Chow Rings ◽

Geometric Methods ◽

The Right

This book provides an introduction to algebraic cycles on complex algebraic varieties, to the major conjectures relating them to cohomology, and even more precisely to Hodge structures on cohomology. The book is intended for both students and researchers, and not only presents a survey of the geometric methods developed in the last thirty years to understand the famous Bloch-Beilinson conjectures, but also examines recent work by the author. It focuses on two central objects: the diagonal of a variety—and the partial Bloch-Srinivas type decompositions it may have depending on the size of Chow groups—as well as its small diagonal, which is the right object to consider in order to understand the ring structure on Chow groups and cohomology. An exploration of a sampling of recent works by the author looks at the relation, conjectured in general by Bloch and Beilinson, between the coniveau of general complete intersections and their Chow groups and a very particular property satisfied by the Chow ring of K3 surfaces and conjecturally by hyper-Kähler manifolds. In particular, the book delves into arguments originating in Nori's work that have been further developed by others.

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