### Stable affine models for algebraic group actions.

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Let $G\to GL\left(V\right)$ be a representation of a reductive linear algebraic group $G$ on a finite-dimensional vector space $V$, defined over an algebraically closed field of characteristic zero. The categorical quotient $X=V//G$ carries a natural stratification, due to D. Luna. This paper addresses the following questions: (i) Is the Luna stratification of $X$ intrinsic? That is, does every automorphism of $V//G$ map each stratum to another stratum? (ii) Are the individual Luna strata in $X$ intrinsic? That is,...

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 ${\mathbb{P}}^{n-1}$, 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...

Let $F$ be a field and $Gr\left(i,{F}^{n}\right)$ be the Grassmannian of $i$-dimensional linear subspaces of ${F}^{n}$. A map $f:Gr\left(i,{F}^{n}\right)\to Gr\left(j,{F}^{n}\right)$ is called nesting if $l\subset f\left(l\right)$ for every $l\in Gr\left(i,{F}^{n}\right)$. Glover, Homer and Stong showed that there are no continuous nesting maps $Gr\left(i,{\mathbb{C}}^{n}\right)\to Gr\left(j,{\mathbb{C}}^{n}\right)$ except for a few obvious ones. We prove a similar result for algebraic nesting maps $Gr\left(i,{F}^{n}\right)\to Gr\left(j,{F}^{n}\right)$, where $F$ is an algebraically closed field of arbitrary characteristic. For $i=1$ this yields a description of the algebraic sub-bundles of the tangent bundle to the projective space ${P}_{F}^{n}$.

In this paper we consider questions of the following type. Let $k$ be a base field and $K/k$ be a field extension. Given a geometric object $X$ over a field $K$ (e.g. a smooth curve of genus $g$), what is the least transcendence degree of a field of definition of $X$ over the base field $k$? In other words, how many independent parameters are needed to define $X$? To study these questions we introduce a notion of essential dimension for an algebraic stack. Using the resulting theory, we give a complete answer to...

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