Hassett-Tschinkel correspondence: Modality and projective hypersurfaces
type: Item
instance of: scholarly article
title: Hassett-Tschinkel correspondence: Modality and projective hypersurfaces
zbMATH Open document ID: 1248.14053
author: Ivan V. Arzhantsev
author: Elena V. Sharoyko
published in: Journal of Algebra
publication date: 2012-04-30T00:00:00Z
full work available at URL: https://arxiv.org/abs/0912.1474
reviewed by: Dmitri I. Panyushev
Mathematics Subject Classification ID: 14L30
Mathematics Subject Classification ID: 14M17
Mathematics Subject Classification ID: 13H99
zbMATH DE Number: 6029091
zbMATH Keywords: unipotent group
zbMATH Keywords: local algebra
zbMATH Keywords: modality
zbMATH Keywords: projective quadric
MaRDI profile type: Publication
OpenAlex ID: W2962998283
arXiv ID: 0912.1474
cites work: Polyhedral divisors and algebraic torus actions
cites work: Gluing affine torus actions via divisorial fans
cites work: Q3217281
cites work: Flag varieties as equivariant compactifications of $\mathbb{G}_{a}^{n}$
cites work: Pairs of commuting nilpotent matrices, and Hilbert function
cites work: Lifting of morphisms to quotient presentations
cites work: Torus actions and combinatorics of polytopes
cites work: Singular del Pezzo surfaces that are equivariant compactifications
cites work: Q4317713
cites work: \({\mathbb{G}_a^M}\) degeneration of flag varieties
cites work: Introduction to Toric Varieties. (AM-131)
cites work: Q4143433
cites work: Q4937714
cites work: Q4744461
cites work: Punctual Hilbert schemes
cites work: Q3209241
cites work: Q3836535
cites work: Plongements d'espaces homogènes
cites work: Generic finite schemes and Hochschild cocycles
cites work: Hassett-Tschinkel correspondence and automorphisms of the quadric
cites work: Q3416497
cites work: Homogeneous spaces and equivariant embeddings
DOI: 10.1016/J.JALGEBRA.2011.09.026
Recommended article: Hassett-Tschinkel correspondence and automorphisms of the quadric
Recommended article: Additive Actions on Projective Hypersurfaces
Recommended article: A general homological Kleiman-Bertini theorem
Recommended article: Additive actions on toric varieties
Recommended article: Group actions on central simple algebras: a geometric approach.
Recommended article: Proper triangular \(\mathbb G_a\)-actions on \(\mathbb A^4\) are translations
Recommended article: Linear \(G_a\) actions on affine spaces and associated rings of invariants
Recommended article: Observability in invariant theory II: Divisors and rational invariants
Recommended article: DECOMPOSITIONS OF Pn INDUCED BY ALGEBRAIC K+- ACTIONS
Recommended article: Products of conjugacy classes in algebraic groups. II
review text: Let \(\mathbb K\) be an algebraically closed field of characteristic zero and \(\mathbb G_n^a\) an \(n\)-dimensional commutative unipotent algebraic group over \(\mathbb K\). In 1999, Hassett and Tschinkel discovered an interesting correspondence between generically transitive actions of \(\mathbb G_n^a\) on the projective space \(\mathbb P^n\) and finite-dimensional local \(\mathbb K\)-algebras (see [\textit{B. Hassett} and \textit{Y. Tschinkel}, Int. Math. Res. Not. 1999, No. 22, 1211--1230 (1999; Zbl 0966.14033)]).NEWLINENEWLINEThe goal of this paper is develop further the theory of this correspondence. Using the corresponding local algebra, the authors calculate modality (in the sense of Vinberg) of generically transitive \(\mathbb G_n^a\)-actions on projective spaces and classify actions of modality one. If \(n\geq 5\), then there are exactly \(n+1\) generically transitive \(\mathbb G_n^a\)-actions of modality one. The authors also characterize generically transitive \(\mathbb G_n^a\)-actions on projective hypersurfaces of a given degree. In particular, it is shown that families (of isomorphism classes) of generically transitive \(\mathbb G_n^a\)-actions on a degenerate quadric in \(\mathbb P^{n+1}\) admit moduli.
description: scientific article; zbMATH DE number 6029091