Affine cones over smooth cubic surfaces

Cheltsov, Ivan; Park, Jihun; Won, Joonyeong

doi:10.4171/jems/622

Cited by 48 publications

(74 citation statements)

References 34 publications

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“…For smooth del Pezzo surfaces of degrees less than or equal to 3, the non-existence of G a -actions on affine cones by their anticanonical divisors was proved in [5,10]. In [6], the existence and non-existence of G a -actions on affine cones over anticanonically polarized del Pezzo surfaces with du Val singularities were fully established according to their singularities and degrees.…”

Section: Theorem 11 Let X Be An Irreducible Affine Algebraic Varietymentioning

confidence: 95%

Flexible affine cones over del Pezzo surfaces of degree 4

Park

Won

2015

European Journal of Mathematics

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Section: Theorem 11 Let X Be An Irreducible Affine Algebraic Varietymentioning

confidence: 95%

Flexible affine cones over del Pezzo surfaces of degree 4

Park

Won

2015

European Journal of Mathematics

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“…Example Consider the affine cone X in

{double-struckK}^{4}

given by the equation

x_{1}^{3} + x_{2}^{3} + x_{3}^{3} + x_{4}^{3} = 0 .

It is proved in that X is rigid. Let

{double-struckK}^{\times}

be the one‐dimensional torus acting on X by scalar multiplication and Z be the projectivization of X in

{double-struckP}^{3}

.…”

Section: Automorphisms Of Rigid Conesmentioning

confidence: 99%

The automorphism group of a rigid affine variety

Arzhantsev

Sergey

2016

Math. Nachr.

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An irreducible algebraic variety $X$ is rigid if it admits no nontrivial action of the additive group of the ground field. We prove that the automorphism group $\text{Aut}(X)$ of a rigid affine variety contains a unique maximal torus $\mathbb{T}$. If the grading on the algebra of regular functions $\mathbb{K}[X]$ defined by the action of $\mathbb{T}$ is pointed, the group $\text{Aut}(X)$ is a finite extension of $\mathbb{T}$. As an application, we describe the automorphism group of a rigid trinomial affine hypersurface and find all isomorphisms between such hypersurfaces.Comment: 12 page

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“…. . , X n ]/ X a 1 1 + · · · + X an n is called a Pham-Brieskorn ring. These rings and the corresponding varieties have been studied extensively and from several angles; paper [10] refers to [13] for a survey.…”

Section: Introductionmentioning

confidence: 99%

On the rigidity of certain Pham-Brieskorn rings

Chitayat

Daigle

2020

Journal of Algebra

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Fix a field k of characteristic zero. If a 1 , . . . , a n (n ≥ 3) are positive integers, the integral domain B a1,...,an = k[X 1 , . . . , X n ]/ X a1 1 + · · · + X an n is called a Pham-Brieskorn ring. It is conjectured that if a i ≥ 2 for all i and a i = 2 for at most one i, then B a1,...,an is rigid. (A ring B is said to be rigid if the only locally nilpotent derivation D : B → B is the zero derivation.) We give partial results towards the conjecture.

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Affine cones over smooth cubic surfaces

Abstract: We show that affine cones over smooth cubic surfaces do not admit non-trivial G a -actions.

Cited by 48 publications

References 34 publications

Flexible affine cones over del Pezzo surfaces of degree 4

Flexible affine cones over del Pezzo surfaces of degree 4

The automorphism group of a rigid affine variety

On the rigidity of certain Pham-Brieskorn rings

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