Minimal generation of ideals of algebraic varieties at points with multilinear tangent cones

In this paper, we define the geometric and algebraic tangent cones at infinity of algebraic varieties and establish the following version at infinity of Whitney’s theorem [Local properties of analytic varieties, in Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse) (Princeton University Press, Princeton, N. J., 1965), pp. 205–244; Tangents to an analytic variety, Ann. of Math. 81 (1965) 496–549]: The geometric and algebraic tangent cones at infinity of complex algebraic varieties coincide. The proof of this fact is based on a geometric characterization of the geometric tangent cone at infinity using the global Łojasiewicz inequality with explicit exponents for complex algebraic varieties. Moreover, we show that the tangent cone at infinity of a complex algebraic variety is actually the part at infinity of this variety [G.-M. Greuel and G. Pfister, A Singular Introduction to Commutative Algebra, 2nd extended edn. (Springer, Berlin, 2008)]. We also show that the tangent cone at infinity of a complex algebraic variety can be computed using Gröbner bases.

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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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Singularities of plane complex curves and limits of Kähler metrics with cone singularities. I: Tangent Cones

Complex Manifolds ◽

10.1515/coma-2017-0005 ◽

2017 ◽

Vol 4 (1) ◽

pp. 43-72 ◽

Cited By ~ 1

Author(s):

Martin de Borbon

Keyword(s):

Vector Field ◽

Einstein Metrics ◽

Projective Line ◽

Tangent Cones ◽

Reeb Vector ◽

Reeb Vector Field ◽

Complex Curves ◽

Complex Dimensions ◽

Irregular Case ◽

Kähler Einstein Metrics

Abstract The goal of this article is to provide a construction and classification, in the case of two complex dimensions, of the possible tangent cones at points of limit spaces of non-collapsed sequences of Kähler-Einstein metrics with cone singularities. The proofs and constructions are completely elementary, nevertheless they have an intrinsic beauty. In a few words; tangent cones correspond to spherical metrics with cone singularities in the projective line by means of the Kähler quotient construction with respect to the S1-action generated by the Reeb vector field, except in the irregular case ℂβ₁×ℂβ₂ with β₂/ β₁ ∉ Q.

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