Quotient Morphism Research Articles

Geometrically pure Ga-actions

We propose the definition of an action σ of the additive group scheme Ga on an affine variety Y to be geometrically pure, which ensures the existence of a geometric quotient of Y by the Ga-action σ if Y is normal. Namely there exists the quotient morphism q:Y→X to a normal affine variety X such that the graph morphism Ψ:Ga×Y→Y×XY is an isomorphism (see Theorem 2.2). Geometric pureness of the given Ga-action is the first criterion ever to guarantee the existence of a geometric quotient Y/Ga. As a consequence, we obtain an algebraic characterization of the affine 3-space A3.

Finitely-cosmall Quotients

In this paper, we first define the notion of finitely-cosmall quotient (singly-cosmall quotient) morphisms. Then we give a characterization of this new concept. We show that an epimorphism p:Y→U is a finitely-cosmall quotient (singly-cosmall quotient) if and only if for any right R-module Z any morphism g:Z→Y such that pg is a finitely-copartial isomorphism (singly-copartial isomorphism) from Z to Y with codomain U is a finitely (singly) split epimorphism. We also investigate the relation between pure-cosmall quotient and finitely-cosmall quotient (singly-cosmall quotient) morphisms. We prove that over a right Noetherian ring R, an epimorphism p:Y→U is a pure-cosmall quotient morphism if and only if p is a finitely-cosmall quotient (singly-cosmall quotient) morphism. Moreover, we obtain an example of right minimal morphisms by using finitely-cosmall quotient (singly-cosmall quotient) morphisms.

Non-symplectic automorphisms of order 3, 4, and 6 of K3 surfaces with smooth quotient

Let X be a K3 surface, and g be a primitive non-symplectic automorphism of X of order 3, 4, or 6. We write the quotient space as and the quotient morphism as . In this paper, we study curves on X obtained by pulling back curves on by p. As a result, we show that if is smooth, then is the projective plane or Hirzebruch surfaces of degree , or 12. In addition, we show that if is smooth, then is determined by the set of fixed points of g and the action of g on rational curves on X.

The asymptotic behavior of finite energy symplectic vortices with admissible metrics

Assume $(X, \omega)$ is a compact symplectic manifold with a Hamiltonian compact Lie group action and the zero in the Lie algebra is a regular value of the moment map $\mu$. We prove that a finite energy symplectic vortex exponentially converges to (un)twisted sectors of the symplectic reduction at cylinder ends whose metrics grow up at least cylindrically fast, without assuming the group action on the level set $\mu^{-1}(0)$ is free. It generalizes the corresponding results by Ziltener [23, 24] under the free action assumption. The result of this paper is the first step in setting up the quotient morphism moduli space induced by the authors in [6]. Necessary preparations in understanding the structure of such moduli spaces are also introduced here. The quotient morphism constructed in [6] is a part of the project on the quantum Kirwan morphism by the authors (see [3, 4, 5]).

Equivariant extensions of $\mathbb{G}_a$-torsors over punctured surfaces

Motivated by the study of the structure of algebraic actions of the additive group on affine threefolds X, we consider a special class of such varieties whose algebraic quotient morphisms X -> X//G(a) restrict to principal homogeneous bundles over the complement of a smooth point of the quotient. We establish basic general properties of these varieties and construct families of examples illustrating their rich geometry. In particular, we give a complete classification of a natural subclass consisting of threefolds X endowed with proper G(a)-actions, whose algebraic quotient morphisms pi : X -> X//G(a) are surjective with only isolated degenerate fibers, all isomorphic to the affine plane A(2) when equipped with their reduced structures.

Affine threefolds admitting $$G_a$$ G a -actions

The structures of affine varieties of dimension greater than two can be explored with the help of fibrations by the affine line or plane and quotient morphisms by $$G_a$$ -actions. We consider $$G_a$$ -actions on affine threefolds and discuss the structure and the singularities of the quotient surface as well as the singular fibers of the quotient morphism.

CANONICAL FACTORIZATION OF THE QUOTIENT MORPHISM FOR AN AFFINE G $$ \mathbb{G} $$ a-VARIETY

Working over a ground field of characteristic zero, this paper studies the quotient morphism π : X → Y for an affine $$ \mathbb{G} $$ a-variety X with affine quotient Y. It is shown that the degree modules associated to the Ga-action give a uniquely determined sequence of dominant $$ \mathbb{G} $$ a-equivariant morphisms, $$ X={X}_r\to {X}_{r-1}\to \cdots \to {X}_1\to {X}_0=Y, $$ where Xi is an affine $$ \mathbb{G} $$ a-variety and Xi + 1 → Xi is birational for each i ≥ 1. This is the canonical factorization of 𝜋. We give an algorithm for finding the degree modules associated to the $$ \mathbb{G} $$ a-action, and this yields the canonical factorization of 𝜋. The algorithm is applied to compute the canonical factorization for several examples, including the homogeneous (2, 5) action on $$ {\mathbb{A}}^3 $$ The Freeness Conjecture, introduced in the paper's last section, asserts that, for any $$ \mathbb{G} $$ a-action on $$ X={\mathbb{A}}^3 $$ the polynomial ring k[X] is a free module over $$ k{\left[X\right]}^{{\mathbb{G}}_a} $$ .

AFFINE EXTENSIONS OF PRINCIPAL ADDITIVE BUNDLES OVER A PUNCTURED SURFACE

The aim of this article is to make a first step towards the classification of complex normal affine α -threefolds X. We consider the case where the restriction of the quotient morphism π: X → S to π−1 (S * ), where S * denotes the complement of some regular closed point in S, is a principal α -bundle. The variety SL2 will be of special interest and a source of many examples. It has a natural right α -action such that the quotient morphism SL2 → 2 restricts to a principal α -bundle over the punctured plane .

Invariants and separating morphisms for algebraic group actions

The first part of this paper is a refinement of Winkelmann's work on invariant rings and quotients of algebraic groups actions on affine varieties, where we take a more geometric point of view. We show that the (algebraic) quotient $X/\!/\!G$ given by the possibly not finitely generated ring of invariants is "almost" an algebraic variety, and that the quotient morphism $\pi\colon X \to X/\!/\! G$ has a number of nice properties. One of the main difficulties comes from the fact that the quotient morphism is not necessarily surjective. These general results are then refined for actions of the additive group $\mathbb{G}_a$, where we can say much more. We get a rather explicit description of the so-called plinth variety and of the separating variety, which measures how much orbits are separated by invariants. The most complete results are obtained for representations. We also give a complete and detailed analysis of Roberts' famous example of a an action of $\mathbb{G}_a$ on 7-dimensional affine space with a non-finitely generated ring of invariants.

Quotients by actions of the derived group of a maximal unipotent subgroup

Let $U$ be a maximal unipotent subgroup of a connected semisimple group $G$ and $U'$ the derived group of $U$. If $X$ is an affine $G$-variety, then the algebra of $U'$-invariants, $k[X]^U'$, is finitely generated and the quotient morphism $\pi: X \to X//U'$ is well-defined. In this article, we study properties of such quotient morphisms, e.g. the property that all the fibres of $\pi$ are equidimensional. We also establish an analogue of the Hilbert-Mumford criterion for the null-cones with respect to $U'$-invariants.

Invariants of a maximal unipotent subgroup and equidimensionality

Let U be a maximal unipotent subgroup of a semisimple group G . If G acts on an affine variety X , then it was proved by Hadžiev (1967) that there is a finitely generated k -algebra A such that k [ X ] U ≃ ( k [ X ] ⊗ A ) G . It follows that k [ X ] U is finitely generated. This note contains two contributions to the theory of U -invariants. First, we obtain a relationship between the fibres of the quotient morphisms π U : X → X / / U and π G : X × Spec ( A ) → ( X × Spec ( A ) ) / / G that contain T -fixed points. (Here T ⊂ N G ( U ) is a maximal torus of G .) For X conical, this implies that π U is equidimensional if and only if π G is. Second, we give a criterion of equidimensionality of π U for a class of varieties with a dense G -orbit (the so-called S -varieties of Vinberg and Popov).

Formula omitted]-fibrations on affine threefolds

An A 1 -fibration on an algebraic surface is well understood and plays an important role in the classification of open algebraic surfaces. We consider A 1 -fibrations on higher-dimensional affine algebraic varieties, mostly in the case of affine threefolds and extend the results known in the case of surfaces. Special attention is paid to degenerate fibers of A 1 -fibrations and singularities of the quotient surface by a G a -action. We distinguish the quotient morphism by a G a -action from an A 1 -fibration. G a -actions will give more information on the A 1 -fibration induced by the actions.

Cross-sections, quotients, and representation rings of semisimple algebraic groups

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny \( \tau :\hat{G} \to G \) is bijective; this answers Grothendieck’s question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg’s theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck’s questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T ⇢ G/T where T is a maximal torus of G and W the Weyl group.

Is the function field of a reductive Lie algebra purely transcendental over the field of invariants for the adjoint action?

AbstractLetkbe a field of characteristic zero, letGbe a connected reductive algebraic group overkand let 𝔤 be its Lie algebra. Letk(G), respectively,k(𝔤), be the field ofk-rational functions onG, respectively, 𝔤. The conjugation action of Gon itself induces the adjoint action of Gon 𝔤. We investigate the question whether or not the field extensionsk(G)/k(G)Gandk(𝔤)/k(𝔤)Gare purely transcendental. We show that the answer is the same fork(G)/k(G)Gandk(𝔤)/k(𝔤)G, and reduce the problem to the case whereGis simple. For simple groups we show that the answer is positive ifGis split of typeAnorCn, and negative for groups of other types, except possiblyG2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of Gon itself.

Quotient morphisms, compositions, and Fredholm index

In this paper we define a composition between two quotient morphisms and prove a multiplication formula for the index of the composition of two Fredholm quotient morphisms. Using this formula and the fact that any linear relation can be seen as a quotient morphism, we obtain a multiplication formula for the index of the composition of two Fredholm linear relations.

Algebraic Hamiltonian actions

In this paper we deal with a Hamiltonian action of a reductive algebraic group G on an irreducible normal affine Poisson variety X. We study the quotient morphism $${\mu_{G,X}//G : X//G \rightarrow \mathfrak{g} //G}$$ of the moment map $${\mu_{G,X} : X\rightarrow \mathfrak{g}}$$ . We prove that for a wide class of Hamiltonian actions (including, for example, actions on generically symplectic varieties) all fibers of the morphism μ G,X //G have the same dimension. We also study the “Stein factorization” of μ G,X //G. Namely, let C G,X denote the spectrum of the integral closure of $${\mu_{G,X}^{*}(\mathbb{K}[\mathfrak{g}]^G)}$$ in $${\mathbb{K}(X)^G}$$ . We investigate the structure of the $${\mathfrak{g} //G}$$ -scheme C G,X . Our results partially generalize those obtained by F. Knop for the actions on cotangent bundles and symplectic vector spaces.

A perturbation result for q-open quotient morphisms in normed spaces and applications to linear relations

Using some techniques of perturbation theory for quotient morphisms between Banach spaces we obtain necessary and sufficient conditions for the stability of the topological index of a q-open quotient morphism under small (with respect to the gap topology) perturbations with quotient morphisms. As an application we obtain necessary and sufficient conditions for the stability of the topological index of an open linear relation with closed multivalued part between normed spaces under small perturbations with linear relations.

Another Approach to Connectedness with Respect to a Closure Operator

We introduce a new concept of connectedness with respect to a categorical closure operator. The concept, which is based on using pseudocomplements in subobject semilattices, naturally generalizes the classical connectedness of topological spaces and we show that it also behaves accordingly. Moreover, as the main result, we prove that the connectedness introduced is preserved, under some natural conditions, by inverse images of subobjects under quotient morphisms. An application of this result in digital topology is discussed too.

Divisorial free modules of relative invariants on Krull domains

An effective construction of relative invariants plays an important role in the study of finite reflection groups (e.g., [J.E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge Stud. Adv. Math., vol. 29, Cambridge Univ. Press, Cambridge, 1990]). Using a combinatorial method, R.P. Stanley (cf. [R.P. Stanley, Relative invariants of finite groups generated by pseudo-reflections, J. Algebra 49 (1977) 134–148; Invariants of finite groups and their applications to combinatorics, Bull. Amer. Math. Soc. (N.S.) 1 (1979) 475–511]) generalized such classical result to a criterion for a module Sym ( V ) χ of invariants of G relative to a character χ to be Sym ( V ) G -free of rank one in the case where G is any finite complex subgroup of GL ( V ) . His criterion is useful in invariant theory of finite groups and in combinatorics. In this paper we will study on relative invariants of a group G consisting of automorphisms of a Krull domain R from the view point of a generalized partial result on ramifications in number theory mentioned in [H. Nakajima, Reduced ramification indices of quotient morphisms under torus actions, J. Algebra 242 (2001) 536–549]. We will give a criterion for R χ to be a free R G -module of rank one for a 1-cocycle χ of G in the unit group U ( R ) , and consequently establish a criterion for module of relative invariants of a finite central extension of algebraic tori to be free, in terms of local characters, which is similar to one in [H. Nakajima, Relative invariants of finite groups, J. Algebra 79 (1982) 218–234; R.P. Stanley, Relative invariants of finite groups generated by pseudo-reflections, J. Algebra 49 (1977) 134–148].

Compactifications des espaces de configuration dans les schémas de Hilbert

Let F(X,n)=X n -Δ be the complement of the union Δ of the diagonals in X n , and let U be a quotient (possibly trivial) of F(X,n) by a subgroup of the symmetric group 𝔖 n . In this work, methods to compactify U inside products of Hilbert Schemes are introduced. Our approach generalizes and unifies previous classical constructions by Schubert-Semple, LeBarz-Keel, Kleiman and Cheah. A more detailed geometrical study is done when n≤3. This includes in particular a complete classification, the determination of the smooth models and a description of the quotient morphisms with respect to the natural actions.