69 research outputs found
A result of Hermite and equations of degree 5 and 6
A classical result from 1861 due to Hermite says that every separable
equation of degree 5 can be transformed into an equation of the form x^5 + b
x^3 + c x + d = 0. Later this was generalized to equations of degree 6 by
Joubert. We show that both results can be understood as an explicit analysis of
certain covariants of the symmetric groups S_5 and S_6. In case of degree 5,
the classical invariant theory of binary forms of degree 5 comes into play
whereas in degree 6 the existence of an outer automorphism of S_6 plays an
essential role.Comment: 14 page
Families of Group Actions, Generic Isotriviality, and Linearization
We study families of reductive group actions on A2 parametrized by curves and show that every faithful action of a non-finite reductive group on A3 is linearizable, i.e. G-isomorphic to a representation of G. The difficulties arise for non-connected groups G. We prove a Generic Equivalence Theorem which says that two affine mor- phisms p: S â Y and q: T â Y of varieties with isomorphic (closed) fibers become isomorphic under a dominant Ěetale base change Ď : U â Y . A special case is the following result. Call a morphism Ď: X â Y a fibration with fiber F if Ď is flat and all fibers are (reduced and) isomorphic to F. Then an affine fibration with fiber F admits an Ěetale dominant morphism Îź: U â Y such that the pull-back is a trivial fiber bundle: U ĂY X â U Ă F . As an application we give short proofs of the following two (known) results: (a) Every affine A1-fibration over a normal variety is locally trivial in the Zariski-topology; (b) Every affine A2-fibration over a smooth curve is locally trivial in the Zariski-topology
Automorphisms of the affine Cremona group
We show that every automorphism of the group Gn := Aut(An) of polynomial automorphisms of complex affine n-space An = Cn is inner up to field automorphisms when restricted to the subgroup TGn of tame automorphisms. This generalizes a result of Julie Deserti who proved this in dimension n = 2 where all automorphisms are tame: TG2 = G2. The methods are different, based on arguments from algebraic group actions
Degree bounds for separating invariants
If V is a representation of a linear algebraic group G, a set S of
G-invariant regular functions on V is called separating if the following holds:
If two elements v,v' from V can be separated by an invariant function, then
there is an f from S such that f(v) is different from f(v'). It is known that
there always exist finite separating sets. Moreover, if the group G is finite,
then the invariant functions of degree <= |G| form a separating set. We show
that for a non-finite linear algebraic group G such an upper bound for the
degrees of a separating set does not exist. If G is finite, we define b(G) to
be the minimal number d such that for every G-module V there is a separating
set of degree less or equal to d. We show that for a subgroup H of G we have
b(H) <= b(G) <= [G:H] b(H) in case H is normal.
Moreover, we calculate b(G) for some specific finite groups.Comment: 11 page
Covariants, Invariant Subsets, and First Integrals
Let be an algebraically closed field of characteristic 0, and let be
a finite-dimensional vector space. Let be the semigroup of all
polynomial endomorphisms of . Let be a subset of which is a
linear subspace and also a semi-subgroup. Both and are
ind-varieties which act on in the obvious way. In this paper, we study
important aspects of such actions. We assign to a linear subspace
of the vector fields on . A subvariety of is said to
-invariant if is in the tangent space of for all in and
in . We show that is -invariant if and only if it is the
union of -orbits. For such , we define first integrals and construct a
quotient space for the -action. An important case occurs when is an
algebraic subgroup of ) and consists of the -equivariant
polynomial endomorphisms. In this case, the associated is the space the
-invariant vector fields. A significant question here is whether there are
non-constant -invariant first integrals on . As examples, we study the
adjoint representation, orbit closures of highest weight vectors, and
representations of the additive group. We also look at finite-dimensional
irreducible representations of SL2 and its nullcone
Varieties Characterized by their Endomorphisms
We show that two varieties X and Y with isomorphic endomorphism semigroups
are isomorphic up to field automorphism if one of them is affine and contains a
copy of the affine line. A holomorphic version of this result is due to the
first author.Comment: 8 page
- âŚ