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Is the Luna stratification intrinsic?

Jochen KuttlerZinovy Reichstein — 2008

Annales de l’institut Fourier

Let G GL ( V ) 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,...

A genericity theorem for algebraic stacks and essential dimension of hypersurfaces

Zinovy ReichsteinAngelo 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...

Essential dimension of moduli of curves and other algebraic stacks

Patrick BrosnanZinovy ReichsteinAngelo Vistoli — 2011

Journal of the European Mathematical Society

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...

Nesting maps of Grassmannians

Corrado De ConciniZinovy Reichstein — 2004

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Let F be a field and G r i , F n be the Grassmannian of i -dimensional linear subspaces of F n . A map f : G r i , F n G r j , F n is called nesting if l f l for every l G r i , F n . Glover, Homer and Stong showed that there are no continuous nesting maps G r i , C n G r j , C n except for a few obvious ones. We prove a similar result for algebraic nesting maps G r i , F n G r j , F n , 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 .

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