Additive actions on hyperquadrics of corank two

Yingqi Liu; Yingqi Liu

doi:10.3934/era.2022001

Electronic Research Archive

2022, Volume 30, Issue 1: 1-34. doi: 10.3934/era.2022001

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Research article

Additive actions on hyperquadrics of corank two

Yingqi Liu ^,

Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

Received: 24 March 2021 Revised: 30 September 2021 Accepted: 12 October 2021 Published: 08 December 2021

For a projective variety $ X $ in $ {\mathbb{P}}^{m} $ of dimension $ n $, an additive action on $ X $ is an effective action of $ {\mathbb{G}}_{a}^{n} $ on $ {\mathbb{P}}^{m} $ such that $ X $ is $ {\mathbb{G}}_{a}^{n} $-invariant and the induced action on $ X $ has an open orbit. Arzhantsev and Popovskiy have classified additive actions on hyperquadrics of corank 0 or 1. In this paper, we give the classification of additive actions on hyperquadrics of corank 2 whose singularities are not fixed by the $ {\mathbb{G}}_{a}^{n} $-action.
- commutative unipotent group action,
- equivariant compactification,
- hyperquadric of corank two,
- symmetric bilinear form,
- simultaneous orthogonal similarity
Citation: Yingqi Liu. Additive actions on hyperquadrics of corank two[J]. Electronic Research Archive, 2022, 30(1): 1-34. doi: 10.3934/era.2022001

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Abstract

For a projective variety $ X $ in $ {\mathbb{P}}^{m} $ of dimension $ n $, an additive action on $ X $ is an effective action of $ {\mathbb{G}}_{a}^{n} $ on $ {\mathbb{P}}^{m} $ such that $ X $ is $ {\mathbb{G}}_{a}^{n} $-invariant and the induced action on $ X $ has an open orbit. Arzhantsev and Popovskiy have classified additive actions on hyperquadrics of corank 0 or 1. In this paper, we give the classification of additive actions on hyperquadrics of corank 2 whose singularities are not fixed by the $ {\mathbb{G}}_{a}^{n} $-action.

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