Tangent Cones

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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Reflexive sheaves, Hermitian–Yang–Mills connections, and tangent cones

Inventiones mathematicae ◽

10.1007/s00222-020-01027-9 ◽

2021 ◽

Author(s):

Xuemiao Chen ◽

Song Sun

Keyword(s):

Tangent Cones ◽

Yang Mills

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Reduced Tangent Cones and Conductor at Multiplanar Isolated Singularities

Communications in Algebra ◽

10.1080/00927870802110367 ◽

2008 ◽

Vol 36 (8) ◽

pp. 2969-2978 ◽

Cited By ~ 3

Author(s):

Alessandro De Paris ◽

Ferruccio Orecchia

Keyword(s):

Tangent Cones ◽

Isolated Singularities

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Complete Intersection Monomial Curves and the Cohen—Macaulayness of Their Tangent Cones

Algebra Colloquium ◽

10.1142/s1005386719000464 ◽

2019 ◽

Vol 26 (04) ◽

pp. 629-642

Author(s):

Anargyros Katsabekis

Keyword(s):

Complete Intersection ◽

Tangent Cone ◽

Affine Space ◽

Intersection Property ◽

Tangent Cones ◽

Monomial Curves ◽

Monomial Curve

Let C(n) be a complete intersection monomial curve in the 4-dimensional affine space. In this paper we study the complete intersection property of the monomial curve C(n + wv), where w > 0 is an integer and v ∈ ℕ4. In addition, we investigate the Cohen–Macaulayness of the tangent cone of C(n + wv).

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On the Cohen-Macaulay Property for Quadratic Tangent Cones

The Electronic Journal of Combinatorics ◽

10.37236/5793 ◽

2016 ◽

Vol 23 (3) ◽

Author(s):

Dumitru I. Stamate

Keyword(s):

Tangent Cone ◽

Gröbner Basis ◽

Numerical Semigroup ◽

Tangent Cones ◽

Grobner Basis

Let $H$ be an $n$-generated numerical semigroup such that its tangent cone $\operatorname{gr}_\mathfrak{m} K[H]$ is defined by quadratic relations. We show that if $n<5$ then $\operatorname{gr}_\mathfrak{m} K[H]$ is Cohen-Macaulay, and for $n=5$ we explicitly describe the semigroups $H$ such that $\operatorname{gr}_\mathfrak{m} K[H]$ is not Cohen-Macaulay. As an application we show that if the field $K$ is algebraically closed and of characteristic different from two, and $n\leq 5$ then $\operatorname{gr}_\mathfrak{m} K[H]$ is Koszul if and only if (possibly after a change of coordinates) its defining ideal has a quadratic Gröbner basis.

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