Additive Actions on Toric Projective Hypersurfaces

Anton, Shafarevich,

doi:10.1007/s00025-021-01462-x

Cited by 6 publications

(6 citation statements)

References 12 publications

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“…So the question is when such a polytope is inscribed in a rectangle. In [99], Proposition 1, all such polytopes were found. It turns out that, apart from the projective space, there are two toric projective hypersurfaces admitting an additive action in every dimension n ⩾ 2; they are the quadrics of rank 3 and 4 (see [99], Theorem 2).…”

Section: Additive Actions On Toric Varietiesmentioning

confidence: 94%

“…Let us mention one more recent result on additive actions on toric varieties. In [99] Shafarevich classified toric projective hypersurfaces admitting an additive action. Every toric hypersurface of dimension n can be represented by a lattice polytope in the lattice M such that the number of lattice points inside this polytope is n + 2.…”

Section: Additive Actions On Toric Varietiesmentioning

confidence: 99%

“…Using the results of [36] Shafarevich computed the automorphism groups of these quadrics. Then he applied the correspondence between additive actions on projective hypersurfaces and pairs (A, U ), where A is a local algebra and U is a hyperplane in the maximal ideal of A that generates A (see Theorem 2.6), and found in [99],…”

Section: Additive Actions On Toric Varietiesmentioning

confidence: 99%

“…Remark 2.28. In [99], § 5, a classification of additive actions on quadrics of small dimension was presented. It was proved that the surface Q(2, 1) admits two additive actions (cf.…”

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confidence: 99%

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Equivariant completions of affine spaces

Arzhantsev¹,

Zaitseva²

2022

Russian Math. Surveys

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We survey recent results on open embeddings of the affine space $\mathbb{C}^n$ into a complete algebraic variety $X$ such that the action of the vector group $\mathbb{G}_a^n$ on $\mathbb{C}^n$ by translations extends to an action of $\mathbb{G}_a^n$ on $X$. We begin with the Hassett-Tschinkel correspondence describing equivariant embeddings of $\mathbb{C}^n$ into projective spaces and present its generalization for embeddings into projective hypersurfaces. Further sections deal with embeddings into flag varieties and their degenerations, complete toric varieties, and Fano varieties of certain types. Bibliography: 109 titles.

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Section: Additive Actions On Toric Varietiesmentioning

confidence: 94%

Section: Additive Actions On Toric Varietiesmentioning

confidence: 99%

Section: Additive Actions On Toric Varietiesmentioning

confidence: 99%

“…Remark 2.28. In [99], § 5, a classification of additive actions on quadrics of small dimension was presented. It was proved that the surface Q(2, 1) admits two additive actions (cf.…”

mentioning

confidence: 99%

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Equivariant completions of affine spaces

Arzhantsev¹,

Zaitseva²

2022

Russian Math. Surveys

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“…There is a description of complete toric varieties with a unique additive action [13]. In [21] projective toric hypersurfaces with additive actions are classified.…”

Section: Introductionmentioning

confidence: 99%

Euler-symmetric projective toric varieties and additive actions

Anton¹

2021

Preprint

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Let G a be the additive group of the field of complex numbers C. We say that an irreducible algebraic variety X of dimension n admits an additive action if there is a regular action of the group G n a = G a × . . . × G a (n times) on X with an open orbit. In 2017 Baohua Fu and Jun-Muk Hwang introduced a class of Euler-symmetric varieties. They gave a classification of Euler-symmetric varieties and proved that any Euler-symmetric variety admits an additive action. In this paper we show that in the case of projective toric varieites the converse is also true. More precisely, a projective toric variety admitting an additive action is an Euler-symmetric variety with respect to any linearly normal embedding into a projective space. Also we discuss some properties of Euler-symmetric projective toric varieties.

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Gorenstein Algebras and Uniqueness of Additive Actions

Beldiev

2023

Results Math

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Additive Actions on Toric Projective Hypersurfaces

Cited by 6 publications

References 12 publications

Equivariant completions of affine spaces

Equivariant completions of affine spaces

Euler-symmetric projective toric varieties and additive actions

Gorenstein Algebras and Uniqueness of Additive Actions

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