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Research area 01 - MATHEMATICS, INFORMATICS, AND SYSTEM SCIENCES, 01-106 - Algebraic geometry

Keywordsalgebraic variety, algebraic group, additive action, unipotent group, Gorenstein algebra

Annotation
An additive action on an algebraic variety is an effective action of commutative unipotent group with an open orbit. In other words, we consider an equivariant completion of affine space, i.e. an affine space as a vector group is embedded as an open subset in an algebraic variety in such a way that the action of the vector group by left translations can be extended to a regular action on entire variety. This case is in some sence opposite to the most developed cases actions of torus and other reductive groups (toric varieties, reductive monoid, spherical varieties,embeddings of homogeneous spaces of complexity one). In the frame of the work under the project “Additive actions on complete algebraic varieties and their generalizations”, supported by RSF grant 19-11-00172, which is succesfuly finished at the moment, we prepared a detailed survey on additive actions on complete varieties. At the moment the survey (I.Arzhantsev and Yu.Zaitseva. Equivariant completions of affine spaces, 68 pages) is finished, inpress and will published in the 4th issue of Uspekhi Mat. Nauk this year (arxiv:2008.09828). The second chapter of the survey contains original results, generalizing the Hassett-Tschinkel correspondence to the case of additive actions on projective hypersurfaces. In particular, we proved that non-degenerate projective hypersurfaces admitting an additive action correspond to Gorenstein local Artinian algebras. This result allowed to generalize Sharoiko's theorem on the uniqueness of additive action on smooth projective quadric. It appears that any non-degenerate projective hypersurface admits at most one additive action (Theorem 2.32). We are sure that this is just the first result in the series that can be obtained on this way, and this project is fully devoted to the development of this topic. Among a variety of structural and classification results on local Artinian algebras the case of Gorenstein algebras has a particular role. In the framework of this project we plan to investigate known facts about Gorenstein algebras and interpret them in terms of additive actions on non-degenerate projective hypersurfaces. It is reasonable to await that even reformulation of known facts from one language to another may lead to interesting results, and the further development of Hassett-Tschinkel technique will allow get a progress both in the investigation of additive actions and structural theory of Gorenstein algebras.

Expected results
In the framework of work under the grant the group plans to deal with some concrete problems on additive actions on projective hypersurfaces to develop the Hassett-Tschinkel technique. We plan to establish a connection between induced and arbitrary additive actions on projective hypersurfaces, in particular, answer a question whether it is possible to obtain any non-induced additive action on projective subvariety as a projection along the subspace for induced additive action on linearly normal embedding of this variety in a bigger projective space. Investigated generalization of Hassett-Tschinkel correspondence on additive actions on projective hypersurfaces will be explicited for the study of properties of hypersurfaces admitting an additive action. For example, it is planned to establish whether the hypersurface depends on the choice of a subspace in the maximal ideal of the corresponding local algebra. An interesting problem is to study the equations that give projective subvarieties and hypersurfaces admitting an additive action. In particular, our group plans to find canonical equations for such hypersurfaces in general or some concrete particular cases. First we are going to investigate extremal cases (hypersurfaces of high and low degrees, non-degenerate hypersurfaces, hypersurfaces with a big kernel of the corresponding multilinear form). We also plan to study the construction reducing an arbitrary pair of a local finite-dimensional algebra and the subspace of its maximal ideal to the extremal case of non-degenerate hypersurface and Gorenstein algebra – for this we use the factorization by the maximal ideal of the algebra containing in the subspace. All of the above problems and the expected results belong to relevant sections of algebraic geometry and the theory of groups of transformations. Advances in the declared directions should lead to significant progress in the understanding of the structure of the mathematical objects under study, which in turn will contribute to solving applied problems and expanding the computational capabilities of modern mathematical science.

Annotation of the results obtained in 2023
Our project aims to study additive actions on algebraic varieties (over an algebraically closed field of characteristic zero), in particular on projective hypersurfaces. An additive action on an algebraic variety is an effective regular action of a commutative unipotent algebraic group with an open orbit. In the case when the algebraic variety is a closed subvariety of a projective space, we are especially interested in so-called induced additive actions, i.e. additive actions that can be extended to regular actions on the ambient projective space. One of the main tools is the generalized Hassett-Tschinkel correspondence establishing a natural bijection between equivalence classes of induced additive actions on an m-dimensional closed subvariety of the n-dimensional projective space and isomorphism classes of so-called H-pairs (A, U), where A is a local finite-dimensional commutative associative unital algebra of dimension n + 1 and U is a subspace in the maximal ideal generating A as a unital algebra. In 2023, one article was published in a journal, and one preprint was prepared and submitted to a journal under the grant support. In https://doi.org/10.1007/s00025-023-01972-w, the following results were obtained. We proved the conjecture of I. Arzhantsev and Y. Zaitseva that there exist at least two non-equivalent induced additive actions on any degenerate hypersurface admitting an induced additive action. This completes the proof of the theorem about equivalence of uniqueness of an induced additive action on a hypersurface and degeneracy of this hypersurface. For a given local finite-dimensional algebra, we provided an explicit construction of two non-isomorphic algebras that are sent to the initial algebra under the reduction with respect to some ideal. It is proved that there exist non-degenerate hypersurfaces of all degrees from 2 to n admitting an additive action in the n-dimensional projective space. The proof consists in an explicit construction of H-pairs; the corresponding Gorenstein algebras are explicitly described in terms of generators and relations. Extremal cases of hypersurfaces of degrees 2 and n were studied; it is proved that, up to isomorphism, there is a unique Gorenstein algebra corresponding to such a hypersurface. For these algebras, it was shown that the corresponding hypersurface does not depend on the choice of a subspace generating the algebra. The question of normality of hypersurfaces coming from the generalized Hassett-Tschinkel correspondence was studied on concrete examples of Gorenstein algebras. In the preprint https://arxiv.org/abs/2308.12096, we considered the relations between non-equivalent additive actions on projective spaces. It was known previously that any two such actions can be conjugated by a birational automorphism of the projective space. We proved a more precise version of this result. Namely, we showed that a conjugating automorphism can be chosen in the affine Cremona group. In addition, this element was interpreted in terms of the generalized Hassett-Tschinkel correspondence. For an H-pair (A, U), we proved that the automorphism defined by the so-called basic polynomials of the algebra A conjugates the corresponding additive action with the standard affine action. In other words, the automorphism is described in terms of basic polynomials. The normality of nondegenerate hypersurfaces corresponding to Young diagrams was studied. The work is preparing for publication. In 2023, the article Beldiev, I. Gorenstein Algebras and Uniqueness of Additive Actions. Results Math 78, 192 (2023). https://doi.org/10.1007/s00025-023-01972-w was published and the preprint Ivan Arzhantsev. On conjugacy of additive actions in the affine Cremona group. https://arxiv.org/abs/2308.12096, 6 pages, was prepared under the grant support. The results were presented at the conference “Lie algebras, algebraic groups and invariant theory”, Moscow, 28.01-02.02.2023, in the poster presentation “Gorenstein algebras and uniqueness of additive actions” by I. Beldiev and also in the talk “Non-uniqueness of an induced additive action on degenerate hypersurfaces” on the seminar of the laboratory of algebraic transformations groups, HSE, 19.04.2023. I. Arzhantsev gave a talk “Equivariant completions of affine space” at the Beijing-Moscow Mathematics Colloquim, HSE, 01.12.2023. Y. Zaitseva gave a talk "Additive actions on projective hypersurfaces" at the seminar of the Laboratory of Algebraic Geometry and its Applications (HSE, Faculty of Mathematics) 17.11.2023.

Publications

1. Beldiev I. Gorenstein Algebras and Uniqueness of Additive Actions Results in Mathematics, 78, 192 (year - 2023) https://doi.org/10.1007/s00025-023-01972-w