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 Göttingen 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 Göttingen covariants to $n$-ary forms using the classical Clebsch transfer principle.

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 Cohen--Macaulay (even complete intersections). Moreover, we give a generalization of this result for the tame concealed-canonical algebras.

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 blow-ups 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.

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.

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.

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.

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 multiplicity-free. 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 Littlewood--Richardson rule.

A smooth affine surface $X$ defined over the complex field $\C$ is an $\ML_0$ surface if the Makar--Limanov 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.

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 Riemann-Roch 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.

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 well-known 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.

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.

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 skew-symmetric analogues) arise as closures of $P$-orbits in $P^u$. We give a unified invariant-theoretic 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.

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.

We show that every equivariant polynomial automorphism of a $\Theta$-repre\-sen\-ta\-tion and of the reduction of an irreducible $\Theta$-representation is a multiple of the identity.