1. CJM 2016 (vol 68 pp. 541)
 GarciaArmas, Mario

Strongly Incompressible Curves
Let $G$ be a finite group. A faithful $G$variety $X$ is called
strongly incompressible if every dominant $G$equivariant rational
map of $X$ onto another faithful $G$variety $Y$ is birational.
We settle the problem of existence of strongly incompressible
$G$curves for any finite group $G$ and any base field $k$ of
characteristic zero.
Keywords:algebraic curves, group actions, Galois cohomology Categories:14L30, 14E07, 14H37 

2. CJM 2014 (vol 67 pp. 848)
 Köck, Bernhard; Tait, Joseph

Faithfulness of Actions on RiemannRoch Spaces
Given a faithful action of a finite group $G$ on an algebraic
curve~$X$ of genus $g_X\geq 2$, we give explicit criteria for
the induced action of~$G$ on the RiemannRoch space~$H^0(X,\mathcal{O}_X(D))$
to be faithful, where $D$ is a $G$invariant divisor on $X$ of
degree at least~$2g_X2$. This leads to a concise answer to the
question when the action of~$G$ on the space~$H^0(X, \Omega_X^{\otimes
m})$ of global holomorphic polydifferentials of order $m$ is
faithful. If $X$ is hyperelliptic, we furthermore provide an
explicit basis of~$H^0(X, \Omega_X^{\otimes m})$. Finally, we
give applications in deformation theory and in coding theory
and we discuss the analogous problem for the action of~$G$ on
the first homology $H_1(X, \mathbb{Z}/m\mathbb{Z})$ if $X$ is a Riemann surface.
Keywords:faithful action, RiemannRoch space, polydifferential, hyperelliptic curve, equivariant deformation theory, Goppa code, homology Categories:14H30, 30F30, 14L30, 14D15, 11R32 

3. CJM 2012 (vol 66 pp. 3)
 Abdesselam, Abdelmalek; Chipalkatti, Jaydeep

On Hilbert Covariants
Let $F$ denote a binary form of order $d$ over the
complex numbers. If $r$ is a divisor of $d$, then the Hilbert covariant
$\mathcal{H}_{r,d}(F)$ vanishes exactly when $F$ is the perfect power of an
order $r$ form. In geometric terms, the coefficients of $\mathcal{H}$ give
defining equations for the image variety $X$ of an embedding $\mathbf{P}^r
\hookrightarrow \mathbf{P}^d$. In this paper we describe a new construction of
the Hilbert covariant; and simultaneously situate it into a wider class of
covariants called the GÃ¶ttingen covariants, all of which vanish on
$X$. We prove that the ideal generated by the coefficients of $\mathcal{H}$
defines $X$ as a scheme. Finally, we exhibit a generalisation of the
GÃ¶ttingen covariants to $n$ary forms using the classical Clebsch transfer principle.
Keywords:binary forms, covariants, $SL_2$representations Categories:14L30, 13A50 

4. CJM 2012 (vol 64 pp. 1222)
 Bobiński, Grzegorz

Normality of Maximal Orbit Closures for Euclidean Quivers
Let $\Delta$ be an Euclidean quiver. We prove that the closures of
the maximal orbits in the varieties of representations of $\Delta$
are normal and CohenMacaulay (even complete intersections).
Moreover, we give a generalization of this result for the tame
concealedcanonical algebras.
Keywords:normal variety, complete intersection, Euclidean quiver, concealedcanonical algebra Categories:16G20, 14L30 

5. CJM 2011 (vol 64 pp. 409)
 Rainer, Armin

Lifting Quasianalytic Mappings over Invariants
Let $\rho \colon G \to \operatorname{GL}(V)$ be a rational finite dimensional complex representation of a reductive linear
algebraic group $G$, and let $\sigma_1,\dots,\sigma_n$ be a system of generators of the algebra of
invariant polynomials $\mathbb C[V]^G$.
We study the problem of lifting mappings $f\colon \mathbb R^q \supseteq U \to \sigma(V) \subseteq \mathbb C^n$
over the mapping of invariants
$\sigma=(\sigma_1,\dots,\sigma_n) \colon V \to \sigma(V)$. Note that $\sigma(V)$ can be identified with the categorical quotient $V /\!\!/ G$
and its points correspond bijectively to the closed orbits in $V$. We prove that if $f$ belongs to a quasianalytic subclass
$\mathcal C \subseteq C^\infty$ satisfying some mild closedness properties that guarantee resolution of singularities in
$\mathcal C$,
e.g., the real analytic class, then $f$ admits a lift of the
same class $\mathcal C$ after desingularization by local blowups and local power substitutions.
As a consequence we show that $f$ itself allows for a lift
that belongs to $\operatorname{SBV}_{\operatorname{loc}}$, i.e., special functions of bounded variation.
If $\rho$ is a real representation of a compact Lie group, we obtain stronger versions.
Keywords:lifting over invariants, reductive group representation, quasianalytic mappings, desingularization, bounded variation Categories:14L24, 14L30, 20G20, 22E45 

6. CJM 2011 (vol 63 pp. 1058)
 Easton, Robert W.

$S_3$covers of Schemes
We analyze flat $S_3$covers of schemes, attempting to create
structures parallel to those found in the abelian and triple cover
theories. We use an initial local analysis as a guide in finding a
global description.
Keywords:nonabelian groups, permutation group, group covers, schemes Category:14L30 

7. CJM 2009 (vol 62 pp. 262)
 Goresky, Mark; MacPherson, Robert

On the Spectrum of the Equivariant Cohomology Ring
If an algebraic torus $T$ acts on a complex projective algebraic
variety $X$, then the affine scheme $\operatorname{Spec}
H^*_T(X;\mathbb C)$ associated with the equivariant cohomology is
often an arrangement of linear subspaces of the vector space
$H_2^T(X;\mathbb C).$ In many situations the ordinary cohomology ring
of $X$ can be described in terms of this arrangement.
Categories:14L30, 54H15 

8. CJM 2009 (vol 62 pp. 473)
 Yun, Zhiwei

GoreskyâMacPherson Calculus for the Affine Flag Varieties
We use the fixed point arrangement technique developed by
Goresky and MacPherson to calculate the part of the
equivariant cohomology of the affine flag variety $\mathcal{F}\ell_G$ generated
by degree 2. We use this result to show that the vertices of the
moment map image of $\mathcal{F}\ell_G$ lie on a paraboloid.
Categories:14L30, 55N91 

9. CJM 2009 (vol 61 pp. 351)
 Graham, William; Hunziker, Markus

Multiplication of Polynomials on Hermitian Symmetric spaces and LittlewoodRichardson Coefficients
Let $K$ be a complex reductive algebraic group and $V$ a
representation of $K$. Let $S$ denote the ring of polynomials on
$V$. Assume that the action of $K$ on $S$ is multiplicityfree. If
$\lambda$ denotes the isomorphism class of an irreducible
representation of $K$, let $\rho_\lambda\from K \rightarrow
GL(V_{\lambda})$ denote the corresponding irreducible representation
and $S_\lambda$ the $\lambda$isotypic component of $S$. Write
$S_\lambda \cdot S_\mu$ for the subspace of $S$ spanned by products of
$S_\lambda$ and $S_\mu$. If $V_\nu$ occurs as an irreducible
constituent of $V_\lambda\otimes V_\mu$, is it true that
$S_\nu\subseteq S_\lambda\cdot S_\mu$? In this paper, the authors
investigate this question for representations arising in the context
of Hermitian symmetric pairs. It is shown that the answer is yes in
some cases and, using an earlier result of Ruitenburg, that in the
remaining classical cases, the answer is yes provided that a
conjecture of Stanley on the multiplication of Jack polynomials is
true. It is also shown how the conjecture connects multiplication in
the ring $S$ to the usual LittlewoodRichardson rule.
Keywords:Hermitian symmetric spaces, multiplicity free actions, LittlewoodRichardson coefficients, Jack polynomials Categories:14L30, 22E46 

10. CJM 2008 (vol 60 pp. 109)
 Gurjar, R. V.; Masuda, K.; Miyanishi, M.; Russell, P.

Affine Lines on Affine Surfaces and the MakarLimanov Invariant
A smooth affine surface $X$ defined over the complex field $\C$ is an $\ML_0$ surface if the
MakarLimanov invariant $\ML(X)$ is trivial. In this paper we study the topology and geometry of
$\ML_0$ surfaces. Of particular interest is the question: Is every curve $C$ in $X$ which is isomorphic
to
the affine line a fiber component of an $\A^1$fibration
on $X$? We shall show that the answer is affirmative if the Picard number
$\rho(X)=0$, but negative in case $\rho(X) \ge 1$. We shall also study the ascent and descent of
the $\ML_0$ property under proper maps.
Categories:14R20, 14L30 

11. CJM 2003 (vol 55 pp. 693)
 Borne, Niels

Une formule de RiemannRoch Ã©quivariante pour les courbes
Soit $G$ un groupe fini agissant sur une courbe alg\'ebrique
projective et lisse $X$ sur un corps alg\'ebriquement clos $k$. Dans
cet article, on donne une formule de RiemannRoch pour la
caract\'eristique d'Euler \'equivariante d'un $G$faisceau inversible
$\mathcal{L}$, \`a valeurs dans l'anneau $R_k (G)$ des caract\`eres du
groupe $G$. La formule donn\'ee a un bon comportement fonctoriel en
ce sens qu'elle rel\`eve la formule classique le long du morphisme
$\dim \colon R_k (G) \to \mathbb{Z}$, et est valable m\^eme pour une
action sauvage. En guise d'application, on montre comment calculer
explicitement le caract\`ere de l'espace des sections globales d'une
large classe de $G$faisceaux inversibles, en s'attardant sur le cas
particulier d\'elicat du faisceau des diff\'erentielles sur la courbe.
Keywords:group actions on varieties or schemes,, RiemannRoch theorems Categories:14L30, 14C40 

12. CJM 2002 (vol 54 pp. 554)
 Hausen, Jürgen

Equivariant Embeddings into Smooth Toric Varieties
We characterize embeddability of algebraic varieties into smooth toric
varieties and prevarieties. Our embedding results hold also in an
equivariant context and thus generalize a wellknown embedding theorem
of Sumihiro on quasiprojective $G$varieties. The main idea is to
reduce the embedding problem to the affine case. This is done by
constructing equivariant affine conoids, a tool which extends the
concept of an equivariant affine cone over a projective $G$variety to
a more general framework.
Categories:14E25, 14C20, 14L30, 14M25 

13. CJM 2000 (vol 52 pp. 1018)
 Reichstein, Zinovy; Youssin, Boris

Essential Dimensions of Algebraic Groups and a Resolution Theorem for $G$Varieties
Let $G$ be an algebraic group and let $X$ be a generically free $G$variety.
We show that $X$ can be transformed, by a sequence of blowups with smooth
$G$equivariant centers, into a $G$variety $X'$ with the following
property the stabilizer of every point of $X'$ is isomorphic to a
semidirect product $U \sdp A$ of a unipotent group $U$ and a
diagonalizable group $A$.
As an application of this result, we prove new lower bounds on essential
dimensions of some algebraic groups. We also show that certain
polynomials in one variable cannot be simplified by a Tschirnhaus
transformation.
Categories:14L30, 14E15, 14E05, 12E05, 20G10 

14. CJM 1999 (vol 51 pp. 616)
 Panyushev, Dmitri I.

Parabolic Subgroups with Abelian Unipotent Radical as a Testing Site for Invariant Theory
Let $L$ be a simple algebraic group and $P$ a parabolic subgroup
with Abelian unipotent radical $P^u$. Many familiar varieties
(determinantal varieties, their symmetric and skewsymmetric
analogues) arise as closures of $P$orbits in $P^u$. We give a
unified invarianttheoretic treatment of various properties of
these orbit closures. We also describe the closures of the
conormal bundles of these orbits as the irreducible components of
some commuting variety and show that the polynomial algebra
$k[P^u]$ is a free module over the algebra of covariants.
Categories:14L30, 13A50 

15. CJM 1998 (vol 50 pp. 929)
 Broer, Abraham

Decomposition varieties in semisimple Lie algebras
The notion of decompositon class in a semisimple Lie algebra is a
common generalization of nilpotent orbits and the set of
regular semisimple elements. We prove that the closure of a
decomposition class has many properties in common with nilpotent
varieties, \eg, its normalization has rational singularities.
The famous Grothendieck simultaneous resolution is related to the
decomposition class of regular semisimple elements. We study the
properties of the analogous commutative diagrams associated to
an arbitrary decomposition class.
Categories:14L30, 14M17, 15A30, 17B45 

16. CJM 1998 (vol 50 pp. 378)