Локально нильпотентные дифференцирования, аддитивные действия и алгебраические моноиды тема диссертации и автореферата по ВАК РФ 00.00.00, кандидат наук Зайцева Юлия Ивановна

• Зайцева Юлия Ивановна
• кандидат наук
• 2024

Оглавление диссертации кандидат наук Зайцева Юлия Ивановна

Contents

Introduction

1. Locally nilpotent derivations on trinomial algebras

1.1. Preliminaries

1.2. Auxiliary lemmas

1.3. Main results

1.4. Roots of trinomial algebras

2. Affine algebraic monoids

2.1. Commutative algebraic monoids

2.2. Commutative monoids on affine spaces

2.3. Semidirect products and the torie structure

2.4. Classification of affine monoids of corank one

2.5. Idempotent elements

2.6. The set of idempotents and the center

3. Affine homogeneous varieties and suspensions

3.1. Regular suspensions

3.2. The case of surfaces

4. Additive actions on projective hypersurfaces

4.1. Additive actions on projective spaces and projective subvarieties

4.2. The case of projective hypersurfaces: equations

4.3. The case of projective hypersurfaces: invariant multilinear forms

4.4. Xon-degenerate hypersurfaces and Gorenstein algebras 80 Conclusion 84 Publications 85 References 85 Appendix A

Введение диссертации (часть автореферата) на тему «Локально нильпотентные дифференцирования, аддитивные действия и алгебраические моноиды»

Introduction

Description of the research area. We study affine algebraic varieties over an algebraically closed field K of characteristic zero and their regular automorphisms. It is known that the automorphism group Aut(X) of an affine varietv X need not be a (finite-dimensional) algebraic group, and it is an important problem to describe algebraic subgroups of Aut(X). A subgroup G in Aut(X) is said to be algebraic if G has a structure of an algebraic group such that the action G x X ^ X is a morphism of algebraic varieties, A normal algebraic variety is called torie if it admits an action of an algebraic torus with an open orbit. The theory of torie varieties has deep connections with combinatorics, commutative algebra and convex geometry; see for example monographs |Fu93, CLS11|, In particular, any torie variety is given by its fan consisting of rational polyhedral cones, and a lot of geometric questions on torie varieties have answers in combinatorial terms.

Any affine algebraic group G has a unique maximal torus T up to conjugation. If G is connected, then it is generated by its maximal torus T and one-parameter unipotent subgroups Ga = (K, +) normalized by T, which are called root subgroups with respect to T; see [Hu75]. One may apply this to describe the automorphism group Aut(X) of a

X

In his seminal work [De70], Demazure described Aut(X) and introduced special elements in the character lattice of the acting torus, which are in bijeetion with root subgroups, Nowadays, these elements are called Demazure roots.

In |Cox95|, Cox suggested another method to describe the automorphism group of a complete torie variety. He defined an important invariant of a variety called the homogeneous coordinate ring or the Cox ring; see also |Bat93|, In the torie ease, the Cox ring is a graded polynomial ring, and the description of the automorphism group of a complete torie variety can be reduced to the description of Ga-actions on an affine space normalized by the diagonal torus action and centralized by a certain quasitorus.

An important tool to study Ga-actions are locally nilpotent derivations, A K-linear map 5: R ^ R is called a derivation of the algebra R if 5(fg) = 5(f)g + f5(g) for any f,g e R. A derivation 5 is said to be locally nilpotent if for any f e R there exists k e Z>0 such that 5k(f) = 0 If R is graded by some abelian group, then a derivation 5 of R is

R

X

functions K[X] by the character lattice of the acting torus. In turn, regular actions of one-parameter unipotent subgroups Ga on X are in bijeetion with locally nilpotent derivations on K[X], Further, a Ga-action is normalized by a torus if and only if the corresponding locally nilpotent derivation is homogeneous with respect to the grading defined by this torus. This technique is used in many works in order to describe automorphisms and to study the geometry of affine varieties, see e.g. |FZ05, AH06, LilO-2, AKZ12, AG17, Shl7, Arl8, CS19, Ca21, LRU22|, Moreover, lifting automorphisms to the spectrum of the Cox ring, one can reduce the study of automorphisms of certain projective varieties to the study of homogeneous automorphisms of affine varieties equipped with an action of the so-called Xeron-Severi quasitorus, see |Cox95, HKOO, BH03| for the original approach and |Ga08, AGIO, AHHL14, APS14, AK15, ADHL15| for further developments. This method opens a wide area of applications and motivates the study of graded affine algebras and homogeneous locally nilpotent derivations.

Recall that the complexity of a torus action is the codimension of a generic orbit, Torie varieties are precisely varieties with a torus action of complexity zero. Their natural generalization are varieties with torus actions of complexity one. Any afline torie variety is given by binomials, see e.g. |St96, Chapter 4|, At the same time, the study of varieties with torus action of complexity one is related to some specific relations called trinomials,

Bv a trinomial we mean a polynomial of the form g — T00 + T1 + T2 such that each variable appears in at most one monomial Tf, While the Cox ring of a torie variety is a polynomial ring, the Cox ring of a variety with a torus action of complexity one is a factor-algebra of a polynomial ring by an ideal generated by trinomials; see |HS10, HHS11, HH13, AHHL14, HW17|, This motivates us to study homogeneous locally nilpotent derivations on trinomial algebras; see subsection 1) and Section 1,

In parallel to the theory of algebraic groups, the theory of algebraic monoids has been developed. An algebraic variety X with an associative multiplication X x X ^ X is called an algebraic monoid if the multiplication is a morphism of algebraic varieties and has a

X

X

X

ding G ^ X of an affine algebraic group G such that the action of the group G x G by left and right multiplications on G can be extended to the action of G x G on X, It ap-

G

GG

Theorem 1| for characteristic zero and |Ri98, Proposition 1| for the general case. The theory of affine algebraic monoids and group embeddings is a rich area of mathematics lying at the intersection of algebra, algebraic geometry, combinatorics and representation theory; see |Pu88, Vi95, Ri98, Re05| for general presentations.

An affine algebraic monoid is called reductive if its group of invertible elements is a reductive affine algebraic group. The theory of reductive monoids is the most developed, see e.g. the combinatorial classification of reductive monoids in |Vi95, Ri98|, It is based on the representation theory of reductive groups, i.e., the highest weight theory.

The next possible aim is a classification for other classes of monoids, for example, solvable or commutative, A monoid is both reductive and commutative if and only if it is a torie variety with a canonical multiplication. It is important to find all monoid structures on a fixed variety, for example, on an affine space. It is also interesting to obtain explicit formulas for multiplications in monoids; see subsection 2) and Section 2,

Let us focus on affine varieties with big automorphism groups. The most interesting is the transitive case. The classical examples here are homogeneous spaces of affine algebraic groups. It is natural to ask whether there are other varieties with the transitive action of the automorphism group. Such examples can be found among Danielewski surfaces and Danilov-Cizatullin surfaces, see |Ci70, GD77, ML01, Du04|, subsection 3) and Section 3,

Let us recall the notion of flexibility, which is close to that of homogeneity. The subgroup of the automorphism group Aut(X) of a varietv X generated by all Ga-subgroups in Aut(X) is called the special automorphism group SAut(X), A smooth point x of a variety X is called flexible if the tangent space to X at the point x is generated by tangents to orbits of G„-subgroups passing through the point x, A variety X is called flexible if any smooth point X

(b) the group SAut(X) acts on the set of smooth points of X transitively. Moreover, if the variety X has dimension at least 2, then these conditions are equivalent to

(c) the group SAut(X) acts on the set of smooth points of X infinitely transitive.

There are many interesting examples of flexible varieties. One useful construction here is a suspension Susp(X, f) = {uv = f (x)} C A2 x X over an affine variety X If X is a flexible

X

well; see |AKZ12| for an algebraically closed field of characteristic zero and |KM12| for the case of the ground field R, In the context of automorphism groups suspensions were considered for the first time in |KZ99|,

The last subject we are interested in is an additive analogue of torio varieties. The idea is to replace the multiplicative group of the ground field by an additive one and consider a commutative unipotent group G", By an additive action on a variety we mean an action of the group G" with an open orbit. In other words, we consider open equivariant embeddings of vector groups into algebraic varieties. The affine case is trivial here since any orbit of a unipotent group on an affine variety is closed. For projective varieties, the theory is nontrivial even for a projective space. In |HT99|, Hassett and Tsehinkel establish a correspondence between finite-dimensional commutative local unital algebras and additive actions on projective spaces; see also |KL84|, It appears that there are infinite families of pairwise non-equivalent additive actions on P" starting from n = 6,

Similar approach may be applied to the study of additive actions on projective hyper-surfaces, This time we need an additional data: a hvperplane U in the maximal ideal m of the algebra A It is known that the degree of the hvpersurface X corresponding to a pair (A, U) equals the maximal exponent d with md ^ U, see [AS11], An additive action on a non-degenerate quadrie is unique |AS11|, and (infinitely many) induced additive actions on degenerate quadries of corank one are described in |AP14|, In |Bazl3|, the case of cubic hypersurfaees is studied; in particular, it turns out that an induced additive action on a non-degenerate cubic hvpersurface is also unique. The next step is to study both non-degenerate and degenerate hypersurfaees of arbitrary degree; see subsection 4) and Section 4,

Main results. Main results of the thesis are as follows,

1, All homogeneous locally nilpotent derivations on trinomial algebras are elementary,

2, Classifications of commutative monoid structures on A3 and of monoid structures of corank one on an arbitrary normal affine variety,

3, A classification of Danielewski surfaces that are homogeneous varieties but not homogeneous spaces,

4, The uniqueness of an induced additive action on a non-degenerate projective hvpersurface.

Заключение диссертации по теме «Другие cпециальности», Зайцева Юлия Ивановна

Заключение

Мы изучаем аффинные алгебраические многообразия над алгебраически замкнутым нолем характеристики нуль и их регулярные автоморфизмы. Дня этих долой мы используем различные техники, включая градуировки и локально нилыютеитиые дифференцирования на алгебрах регулярных функций. Мы исследуем связанные объекты, такие как аффинные алгебраические моноиды, многообразия с транзитивными действиями групп автоморфизмов и многообразия с действиями некоторых алгебраических групп с открытой орбитой,

В раздело 1 мы описываем однородные локально нилыютеитиые дифференцирования на триномиальных алгебрах. Доказано, что они элементарны. Это позволяет дать критерий существования однородных локально ни.ныютентиых дифференцирований на триномиальных алгебрах, новое доказательство критерия жёсткости факториаль-пых триномиальных гиперповерхностей и найти корни триномиальных гиперповерхностей.

В раздело 2 мы классифицируем структуры коммутативных алгебраических моноидов на трёхмерном аффинном пространстве и структуры моноида коранга один на произвольном нормальном аффинном многообразии. Оказывается, что любое многообразие, допускающее структуру моноида коранга один, является торическим, и в этом случае умножения описываются с помощью корней Демазюра многообразия. Используя этот результат мы получаем описание идемнотонтов и центра моноида и изучаем связь между ними,

В раздело 3 мы приводим некоторые результаты об аффинных однородных многообразиях, то есть аффинных алгебраических многообразиях с транзитивным действием группы автоморфизмов. Мы даём критерий гладкости надстройки и классификацию поверхностей Даниловского, которые являются однородными многообразиями, но не однородными пространствами,

Раздел 4 посвящен изучению аддитивных действий, то есть действий векторной группы с открытой орбитой. Мы доказываем единственность индуцированного аддитивного действия на невырожденной проективной гиперповерхности.

Результаты диссертации могут быть использованы в дальнейших изучениях алгебраических групп преобразований и автоморфизмов многообразий.

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