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Zinovy Reichstein

Publications and source records attributed to Zinovy Reichstein.

At least 55 records · Page 3Linked to original sources
arXiv2016

On the Medvedev-Scanlon Conjecture for Minimal Threefolds of Non-Negative Kodaira Dimension

Motivated by work of Zhang from the early `90s, Medvedev and Scanlon formulated the following conjecture. Let $K$ be an algebraically closed field of characteristic $0$ and let $X$ be a quasiprojective variety defined over $K$ endowed with a dominant rational self-map $Φ$. Then there exists a point $α\in X(K)$ with Zariski dense orbit under $Φ$ if and only if $Φ$ preserves no nontrivial rational fibration, i.e., there exists no non-constant rational function $f\in K(X)$ such that $Φ^*(f)=f$. The Medvedev-Scanlon conjecture holds when $K$ is uncountable. The case where $K$ is countable (e.g., $K=\overline{\mathbb{Q}}$) is much more difficult; here the conjecture has only been proved in a small number of special cases. In this paper we show that the Medvedev-Scanlon conjecture holds for all varieties of positive Kodaira dimension, and explore the case of Kodaira dimension $0$. Our results are most complete in dimension $3$.

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arXiv2016

On the dimension of the locus of determinantal hypersurfaces

The characteristic polynomial of an $r$-tuple $(A_1,..., A_r)$ of $n \times n$ matrices is the determinant $\det(x_0 I + x_1 A_1 + ... + x_r A_r)$. We show that if $r$ is at least 3 and $A = (A_1,..., A_r)$ is an $r$-tuple of matrices in general position, then up to conjugacy there are only finitely many $r$-tuples of matrices with the same characteristic polynomial as $A$. Equivalently, the locus of determinantal hypersurfaces of degree $n$ in $P^r$ is irreducible of dimension $(r-1)n^2 + 1$.

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arXiv2015

Essential dimension and error-correcting codes

One of the important open problems in the theory of central simple algebras is to compute the essential dimension of $\operatorname{GL}_n/μ_m$, i.e., the essential dimension of a generic division algebra of degree $n$ and exponent dividing $m$. In this paper we study the essential dimension of groups of the form \[ G=(\operatorname{GL}_{n_1} \times \dots \times \operatorname{GL}_{n_r})/C \, , \] where $C$ is a central subgroup of $\operatorname{GL}_{n_1} \times \dots \times \operatorname{GL}_{n_r}$. Equivalently, we are interested in the essential dimension of a generic $r$-tuple $(A_1, \dots, A_r)$ of central simple algebras such that $\operatorname{deg}(A_i) = n_i$ and the Brauer classes of $A_1, \dots, A_r$ satisfy a system of homogeneous linear equations in the Brauer group. The equations depend on the choice of $C$ via the error-correcting code $\operatorname{Code}(C)$ which we naturally associate to $C$. We focus on the case where $n_1, \dots, n_r$ are powers of the same prime. The upper and lower bounds on $\operatorname{ed}(G)$ we obtain are expressed in terms of coding-theoretic parameters of $\operatorname{Code}(C)$, such as its weight distribution. Surprisingly, for many groups of the above form the essential dimension becomes easier to estimate when $r \geq 3$; in some cases we even compute the exact value. The Appendix by Athena Nguyen contains an explicit description of the Galois cohomology of groups of the form $(\operatorname{GL}_{n_1} \times \dots \times \operatorname{GL}_{n_r})/C$. This description and its corollaries are used throughout the paper.

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arXiv2014

Pseudo-reflection groups and essential dimension

We give a simple formula for the essential dimension of a finite pseudo-reflection group at a prime p and determine the absolute essential dimension for most irreducible pseudo-reflection groups. We also study the "poor man's essential dimension" of an arbitrary finite group, an intermediate notion between the absolute essential dimension and the essential dimension at a prime p.

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arXiv2014

On a dynamical version of a theorem of Rosenlicht

Consider the action of an algebraic group $G$ on an irreducible algebraic variety $X$ all defined over a field $k$. M. Rosenlicht showed that orbits in general position in $X$ can be separated by rational invariants. We prove a dynamical analogue of this theorem, where $G$ is replaced by a semigroup of dominant rational self-maps of $X$. Our semigroup $G$ is not required to have the structure of an algebraic variety and can be of arbitrary cardinality.

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arXiv2014

Joubert's theorem fails in characteristic 2

Let L/K be a separable field extension of degree 6. An 1867 theorem of P. Joubert asserts that if char(K) is different from 2 then L is generated over K by an element whose minimal polynomial is of the form t^6 + a t^4 + b t^2 + ct + d for some a, b, c, d in K. We show that this theorem fails in characteristic 2.

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arXiv2014

A numerical invariant for linear representations of finite groups

We study the notion of essential dimension for a linear representation of a finite group. In characteristic zero we relate it to the canonical dimension of certain products of Weil transfers of generalized Severi-Brauer varieties. We then proceed to compute the canonical dimension of a broad class of varieties of this type, extending earlier results of the first author. As a consequence, we prove analogues of classical theorems of R. Brauer and O. Schilling about the Schur index, where the Schur index of a representation is replaced by its essential dimension. In the last section we show that essential dimension of representations can behave in rather unexpected ways in the modular setting.

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arXiv2013

The Lie algebra of type G_2 is rational over its quotient by the adjoint action

Let G be a split simple group of type G_2 over a field k, and let g be its Lie algebra. Answering a question of Colliot-Thélène, Kunyavskiĭ, Popov, and Reichstein, we show that the function field k(g) is generated by algebraically independent elements over the field of adjoint invariants k(g)^G. Soit G un groupe algébrique simple et déployé de type G_2 sur un corps k. Soit g son algèbre de Lie. On démontre que le corps des fonctions k(g) est transcendant pur sur le corps k(g)^G des invariants adjoints. Ceci répond par l'affirmative à une question posée par Colliot-Thélène, Kunyavskiĭ, Popov et Reichstein.

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arXiv2013

Versality of algebraic group actions and rational points on twisted varieties

We formalize and study several competing notions of versality for an action of a linear algebraic group on an algebraic variety X. Our main result is that these notions of versality are equivalent to various statements concerning rational points on twisted forms of X (existence of rational points, existence of a dense set of rational points, etc.) We give applications of this equivalence in both directions, to study versality of group actions and rational points on algebraic varieties. We obtain similar results on p-versality for a prime integer p. An appendix, containing a letter from J.-P. Serre, puts the notion of versality in a historical perspective.

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arXiv2011

A genericity theorem for algebraic stacks and essential dimension of hypersurfaces

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 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 Genericity Theorem, we prove a new result on the essential dimension of the stack of (not necessarily smooth) local complete intersection curves.

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arXiv2010

Bundles, Cohomology and Truncated Symmetric Polynomials

The cohomology of the classifying space BU(n) of the unitary groups can be identified with the ring of symmetric polynomials on n variables by restricting to the cohomology of BT, where T is a maximal torus in U(n). In this paper we explore the situation where BT = (CP^{infinity})^n is replaced by a product of finite dimensional projective spaces (CP^d)^n, fitting into an associated bundle U(n) x_T (S^{2d+1})^n -> (CP^d)^n -> BU(n). We establish a purely algebraic version of this problem by exhibiting an explicit system of generators for the ideal of truncated symmetric polynomials. We use this algebraic result to give a precise descriptions of the kernel of the homomorphism in cohomology induced by the natural map (CP^d)^n -> BU(n). We also calculate the cohomology of the homotopy fiber of the natural map ES_n x_{S_n} (CP^d)^n -> BU(n).

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arXiv2010

Maximal representation dimension of finite p-groups

The representation dimension of a finite group G is the smallest positive integer m for which there exists an embedding of G in GL_m(C). In this paper we find the largest value of representation dimensions, as Granges over all groups of order p^n, for a fixed prime p and a fixed exponent n greater than zero.

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arXiv2010

Toric-friendly groups

Let G be a connected linear algebraic group over a field k. We say that G is toric-friendly if for any field extension K/k and any maximal K-torus T in G the group G(K) has only one orbit in (G/T)(K). Our main result is a classification of semisimple (and under certain assumptions on k, of connected) toric-friendly groups.

math.AG↗
arXiv2010

Is the function field of a reductive Lie algebra purely transcendental over the field of invariants for the adjoint action?

Let $k$ be a field of characteristic zero, let $G$ be a connected reductive algebraic group over $k$ and let $\mathfrak{g}$ be its Lie algebra. Let $k(G)$, respectively, $k(\mathfrak{g})$, be the field of $k$-rational functions on $G$, respectively, $\mathfrak{g}$. The conjugation action of $G$ on itself induces the adjoint action of $G$ on $\mathfrak{g}$. We investigate the question whether or not the field extensions $k(G)/k(G)^G$ and $k(\mathfrak{g})/k(\mathfrak{g})^G$ are purely transcendental. We show that the answer is the same for $k(G)/k(G)^G$ and $k(\mathfrak{g})/k(\mathfrak{g})^G$, and reduce the problem to the case where $G$ is simple. For simple groups we show that the answer is positive if $G$ is split of type ${\sf A}_{n}$ or ${\sf C}_n$, and negative for groups of other types, except possibly ${\sf G}_{2}$. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of $G$ on itself. The results and methods of this paper have played an important part in recent A. Premet's negative solution (arxiv:0907.2500) of the Gelfand--Kirillov conjecture for finite-dimensional simple Lie algebras of every type, other than ${\sf A}_n$, ${\sf C}_n$, and ${\sf G}_2$.

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arXiv2009

Essential p-dimension of algebraic tori

The essential dimension is a numerical invariant of an algebraic group G which may be thought of as a measure of complexity of G-torsors over fields. A recent theorem of N. Karpenko and A. Merkurjev gives a simple formula for the essential dimension of a finite p-group. We obtain similar formulas for the essential p-dimension of a broader class of groups, which includes all algebraic tori.

math.GR↗
arXiv2009

Essential dimension of moduli of curves and other algebraic stacks

In this paper we address 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. In particular, we give a complete answer to the question above when the geometric objects X are smooth or stable curves, and, in the appendix by N. Fakhruddin, for principally polarized abelian varieties. This paper overlaps with our earlier preprint arXiv:math/0701903 . That preprint has splintered into several parts, which have since acquired a life of their own. In particular, see "Essential dimension, spinor groups, and quadratic forms", by the same authors, and "Some consequences of the Karpenko-Merkurjev theorem", by Meyer and Reichstein (arXiv:0811.2517).

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