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A categorical quotient in the category of dense constructible subsets

Devrim Celik (2009)

Colloquium Mathematicae

A. A'Campo-Neuen and J. Hausen gave an example of an algebraic torus action on an open subset of the affine four space that admits no quotient in the category of algebraic varieties. We show that this example admits a quotient in the category of dense constructible subsets and thereby answer a question of A. Białynicki-Birula.

A combinatorial construction of sets with good quotients by an action of a reductive group

Joanna Święcicka (2001)

Colloquium Mathematicae

The aim of this paper is to construct open sets with good quotients by an action of a reductive group starting with a given family of sets with good quotients. In particular, in the case of a smooth projective variety X with Pic(X) = 𝒵, we show that all open sets with good quotients that embed in a toric variety can be obtained from the family of open sets with projective good quotients. Our method applies in particular to the case of Grassmannians.

A fixed point formula of Lefschetz type in Arakelov geometry II: A residue formula

Kai Köhler, Damien Roessler (2002)

Annales de l’institut Fourier

This is the second of a series of papers dealing with an analog in Arakelov geometry of the holomorphic Lefschetz fixed point formula. We use the main result of the first paper to prove a residue formula "à la Bott" for arithmetic characteristic classes living on arithmetic varieties acted upon by a diagonalisable torus; recent results of Bismut- Goette on the equivariant (Ray-Singer) analytic torsion play a key role in the proof.

A general Hilbert-Mumford criterion

Jürgen Hausen (2003)

Annales de l’institut Fourier

Let a reductive group G act on an algebraic variety X . We give a Hilbert-Mumford type criterion for the construction of open G -invariant subsets V X admitting a good quotient by G .

A genericity theorem for algebraic stacks and essential dimension of hypersurfaces

Zinovy Reichstein, Angelo Vistoli (2013)

Journal of the European Mathematical Society

We compute the essential dimension of the functors Forms n , d and Hypersurf n , d of equivalence classes of homogeneous polynomials in n variables and hypersurfaces in n 1 , respectively, over any base field k of characteristic 0 . Here two polynomials (or hypersurfaces) over K are considered equivalent if they are related by a linear change of coordinates with coefficients in K . Our proof is based on a new Genericity Theorem for algebraic stacks, which is of independent interest. As another application of the...

A Geometrical Construction for the Polynomial Invariants of some Reflection Groups

Sarti, Alessandra (2005)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: Primary 20F55, 13F20; Secondary 14L30.We construct invariant polynomials for the reflection groups [3, 4, 3] and [3, 3, 5] by using some special sets of lines on the quadric P1 × P1 in P3. Then we give a simple proof of the well known fact that the ring of invariants are rationally generated in degree 2,6,8,12 and 2,12,20,30.

A Hilbert-Mumford criterion for SL₂-actions

Jürgen Hausen (2003)

Colloquium Mathematicae

Let the special linear group G : = SL₂ act regularly on a ℚ-factorial variety X. Consider a maximal torus T ⊂ G and its normalizer N ⊂ G. We prove: If U ⊂ X is a maximal open N-invariant subset admitting a good quotient U → U ⃫N with a divisorial quotient space, then the intersection W(U) of all translates g · U is open in X and admits a good quotient W(U) → W(U) ⃫G with a divisorial quotient space. Conversely, we show that every maximal open G-invariant subset W ⊂ X admitting a good quotient W...

A minimal Set of Generators for the Ring of multisymmetric Functions

David Rydh (2007)

Annales de l’institut Fourier

The purpose of this article is to give, for any (commutative) ring A , an explicit minimal set of generators for the ring of multisymmetric functions T S A d ( A [ x 1 , , x r ] ) = A [ x 1 , , x r ] A d 𝔖 d as an A -algebra. In characteristic zero, i.e. when A is a -algebra, a minimal set of generators has been known since the 19th century. A rather small generating set in the general case has also recently been given by Vaccarino but it is not minimal in general. We also give a sharp degree bound on the generators, improving the degree bound previously...

A nonlinearizable action of S 3 on 4

Gene Freudenburg, Lucy Moser-Jauslin (2002)

Annales de l’institut Fourier

The main purpose of this article is to give an explicit algebraic action of the group S 3 of permutations of 3 elements on affine four-dimensional complex space which is not conjugate to a linear action.

A note on semisimple derivations of commutative algebras

Andrzej Tyc (2005)

Colloquium Mathematicae

A concept of a slice of a semisimple derivation is introduced. Moreover, it is shown that a semisimple derivation d of a finitely generated commutative algebra A over an algebraically closed field of characteristic 0 is nothing other than an algebraic action of a torus on Max(A), and, using this, that in some cases the derivation d is linearizable or admits a maximal invariant ideal.

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