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We consider the family of polynomials in $C [x, y, z]$ of the form $x^{2} y - z^{2} - x q (x, z)$ . Two such polynomials $P_{1}$ and $P_{2}$ are equivalent if there is an automorphism $ϕ^{*}$ of $C [x, y, z]$ such that $ϕ^{*} (P_{1}) = P_{2}$ . We give a complete classification of the equivalence classes of these polynomials in the algebraic and analytic category. As a consequence, we find the following results. There are explicit examples of inequivalent polynomials $P_{1}$ and $P_{2}$ such that the zero set of $P_{1} + c$ is isomorphic to the zero set of $P_{2} + c$ for all $c \in C$ . There exist polynomials which are algebraically...

Equations of some wonderful compactifications

Pascal Hivert (2011)

Annales de l’institut Fourier

De Concini and Procesi have defined the wonderful compactification $\bar{X}$ of a symmetric space $X = G / G^{σ}$ where $G$ is a complex semisimple adjoint group and $G^{σ}$ the subgroup of fixed points of $G$ by an involution $σ$ . It is a closed subvariety of a Grassmannian of the Lie algebra $𝔤$ of $G$ . In this paper we prove that, when the rank of $X$ is equal to the rank of $G$ , the variety is defined by linear equations. The set of equations expresses the fact that the invariant alternate trilinear form $w$ on $𝔤$ vanishes on the $(- 1)$ -eigenspace...

Equidimensional actions of algebraic tori

Haruhisa Nakajima (1995)

Annales de l'institut Fourier

Let $X$ be an affine conical factorial variety over an algebraically closed field of characteristic zero. We consider equidimensional and stable algebraic actions of an algebraic torus on $X$ compatible with the conical structure. We show that such actions are cofree and the nullcones of $X$ associated with them are complete intersections.

Equivariant principal bundles for G–actions and G–connections

Indranil Biswas, S. Senthamarai Kannan, D. S. Nagaraj (2015)

Complex Manifolds

Given a complex manifold M equipped with an action of a group G, and a holomorphic principal H–bundle EH on M, we introduce the notion of a connection on EH along the action of G, which is called a G–connection. We show some relationship between the condition that EH admits a G–equivariant structure and the condition that EH admits a (flat) G–connection. The cases of bundles on homogeneous spaces and smooth toric varieties are discussed.

Equivariant vector bundles over affine subsets of the projective line

Corrado De Concini, Fabio Fagnani (1995)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Erratum: Complexity and rank double cones and tensor products decompositions.

D.I. Panyushev (1993)

Commentarii mathematici Helvetici

Espaces homogènes et arithmétique des schémas en groupes réductifs sur les anneaux de Dedekind

Jean-Claude Douai (1995)

Journal de théorie des nombres de Bordeaux

Soit $S$ un schéma arithmétique de dimension $1$ , c’est-à-dire le spectre de l’anneau des entiers d’un corps de nombres ou une courbe algébrique, lisse, irréductible, définie sur un corps fini ou algébriquement clos. Nous associons à un $S$ -espace homogène (à gauche) $X$ d’un groupe réductif $G$ dont l’isotropie est aussi un groupe réductif $H$ une classe caractéristique qui, dans le cas où $H$ est semi-simple, vit dans un $H^{3}$ de $S$ à valeurs dans le noyau du revêtement universel d’une $S$ -forme de $H$ . Cette classe...

Espaces homogenes spheriques.

M. Brion, D. Lima, Th. Vust (1986)

Inventiones mathematicae

Essential dimension: A functorial point of view (after A. Merkurjev).

Berhuy, Grégory, Favi, Giordano (2003)

Documenta Mathematica

Etale $p$ -covers in characteristic $p$

Richard M. Crew (1984)

Compositio Mathematica

Extending algebraic actions.

Arthur G. Wasserman (1999)

Revista Matemática Complutense

There is a well-known procedure -induction- for extending an action of a subgroup H of a Lie group G on a topological space X to an action of G on an associated space. Induction can also extend a smooth action of a subgroup H of a Lie group G on a manifold M to a smooth action of G on an associated manifold. In this paper elementary methods are used to show that induction also works in the category of (nonsingular) real algebraic varieties and regular or entire maps if G is a compact abelian Lie...

Extensions of finite group schemes, and Hopf Galois theory over a complete discrete valuation ring.

C. Greither (1992)

Mathematische Zeitschrift

Factorisation de certains morphismes birationnels

Michel Brion (1994)

Compositio Mathematica

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