Groups with essential dimension one

Essential p-dimension of algebraic groups whose connected component is a torus

Following up on our earlier work and the work of N. Karpenko and A. Merkurjev, we study the essential p-dimension of linear algebraic groups G whose connected component G^0 is a torus.

Finite simple groups of small essential dimension

We discuss the notion of essential dimension of a finite group and explain its relation with birational algebraic geometry. We show how this leads to a (partial) classification of simple finite groups of essential dimension less than or equal to 3.

Essential dimension, stable cohomological dimension, and stable cohomology of finite Heisenberg groups

We compare the notions of essential dimension and stable cohomological dimension of a finite group G, prove that the latter is bounded by the length of any normal series with cyclic quotients for G, and show that, however, this bound is not sharp by showing that the stable cohomological dimension of the finite Heisenberg groups H_p, p any prime, is equal to two.

Essential dimension of the spin groups in characteristic 2

We determine the essential dimension of the spin group Spin(n) as an algebraic group over a field of characteristic 2, for n at least 15. In this range, the essential dimension is the same as in characteristic not 2. In particular, it is exponential in n. This is surprising in that the essential dimension of the orthogonal groups is smaller in characteristic 2. We also find the essential dimension of Spin(n) in characteristic 2 for n at most 10.

The action of the étale fundamental group scheme on the connected component of the essentially finite one

We follow the pattern in a recent paper of Otabe [Ota15] to define an action of the étale fundamental group scheme $π^\text{et}(X)$ on the local component of the essentially finite fundamental group scheme $π^{\mathrm{EF}}(X)$ of Nori. We show that the associated representation is faithful when $X$ is a curve of genus $\geq 2$.

Essential dimension and algebraic stacks

We define and study the essential dimension of an algebraic stack. We compute the essential dimension of the stacks Mgn and MgnBar of smooth, or stable, n-pointed curves of genus g. We also prove a general lower bound for the essential dimension of algebraic groups with a non-trivial center. Using this, we find new exponential lower bounds for the essential dimension of spin groups and new formulas for the essential dimension of some finite p-groups. Finally, we apply the lower bound for spin groups to the theory of the Witt ring of quadratic forms over a field k.

The essential dimension of the normalizer of a maximal torus in the projective linear group

Let p be a prime, k be a field of characteristic different from p containing a primitive p-th root of unity and N be the normalizer of the maximal torus in the projective linear group PGLn. We compute the exact value of the essential dimension ed(N;p) of N at p for every n.

On the essential dimension of infinitesimal group schemes

We discuss essential dimension of group schemes, with particular attention to infinitesimal group schemes. We prove that the essential dimension of a group scheme of finite type over a field k is at least equal to the difference between the dimension of its Lie algebra and its dimension. Furthermore, we show that the essential dimension of a trigonalizable group scheme of length p^{n} over a field of characteristic p>0 is at most n. We give several examples.

Essential dimension of infinitesimal unipotent group schemes

We propose a generalization of Ledet conjecture, which predicts the essential dimension of cyclic $p$-groups in characteristic $p$, for finite commutative unipotent group schemes. And we show some evidence and some consequences of this new conjecture.

Essential dimension of group schemes over a local scheme

In this paper we develop the theory of essential dimension of group schemes over an integral base. Shortly we concentrate over a local base. As a consequence of our theory we give a result of invariance of the essential dimension over a field. The case of group schemes over a discrete valuation ring is discussed. Moreover we propose a generalization of Ledet conjecture, which predicts the essential dimension of cyclic $p$-groups in positive characteristic, for finite commutative unipotent group schemes. And we show some results and some consequences of this new conjecture.

经典型李型分裂有限拟单群的本原 p-维数

The essential p-dimension of the split finite quasi-simple groups of classical Lie type

本文计算了经典型李型分裂有限拟单群在定义素数处的本原 p-维数，涵盖一般线性群、特殊线性群、辛群和正交群产生的拟单群。

Essential Dimensions of A_7 and S_7

We show that Y. Prokhorov's "Simple Finite Subgroups of the Cremona Group of Rank 3" implies that, over any field of characteristic 0, the essential dimensions of the alternating group, A_7, and the symmetric group, S_7, are 4.

群的本质维数与克雷莫纳维数

The essential and Cremona dimensions of a group

本文通过实例支持猜想：有限群在复数域上的克雷莫纳维数不大于其本质维数。

Essential dimension of finite groups in prime characteristic

Given a finite smooth group scheme $G$ over a field of characteristic $p > 0$, we show that the essential dimension of $G$ at $p$ is $0$ when $p$ does not divide the order of $G$, and $1$ when it does.

An upper bound on the essential dimension of a central simple algebra

We prove a new upper bound on the essential p-dimension of the projective linear group PGLn.

通过普遍自由表示计算缩减群的本质维度

Essential dimension of reductive groups via generically free representations

本文通过普遍自由表示提出计算缩减群本质维度的方法，并确定了经典类型和E6类型半单群的本质维度。

李型有限群的本原 $l$ 维数，$l \ eq 2$

The essential $l$-dimension of finite groups of Lie type, $l \neq 2$

本文计算了经典李型有限群在奇素数 $l$ 不等于定义素数时的本原 $l$ 维数，涵盖一般线性群、辛群、正交群、酉群及其单因子。

Application of Multihomogeneous Covariants to the Essential Dimension of Finite Groups

We investigate essential dimension of finite groups over arbitrary fields and give a systematic treatment of multihomogenization, introduced by H.Kraft, G.Schwarz and the author. We generalize the central extension theorem of Buhler and Reichstein and use multihomogenization to substitute and generalize the stack-involved part of the theorem of Karpenko and Merkurjev about the essential dimension of p-groups. One part of this paper is devoted to the study of completely reducible faithful representations. Amongst results concerning faithful representations of minimal dimension there is a comput…

本质实反射群 Specht 理想的偏序集

Posets for Specht ideals of essential real reflection groups

本文研究本质实反射群中 Specht 理想的组合结构，将类型 A、B 的理论推广至类型 D 和二面体群，完成所有无限族的组合分类。

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 of a unipotent group U and a diagonalizable group A. As an application of this and related results, 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.

Essential dimension of double covers of symmetric and alternating groups

I. Schur studied double covers $\widetilde{\Sym}^{\pm}_n$ and $\widetilde{\Alt}_n$ of symmetric groups $\Sym_n$ and alternating groups $\Alt_n$, respectively. Representations of these groups are closely related to projective representations of $\Sym_n$ and $\Alt_n$; there is also a close relationship between these groups and spinor groups. We study the essential dimension $\ed(\widetilde{\Sym}^{\pm}_n)$ and $\ed(\widetilde{\Alt}_n)$. We show that over a base field of characteristic $\neq 2$, $\ed(\widetilde{\Sym}^{\pm}_n)$ and $\ed(\widetilde{\Alt}_n)$ grow exponentially with $n$, similar to $…

Essential dimension of inseparable field extensions

Let k be a base field, K be a field containing k and L/K be a field extension of degree n. The essential dimension ed(L/K) over k is a numerical invariant measuring "the complexity" of L/K. Of particular interest is $τ$(n) = max { ed(L/K) | L/K is a separable extension of degree n}, also known as the essential dimension of the symmetric group $S_n$. The exact value of $τ$(n) is known only for n $\leq$ 7. In this paper we assume that k is a field of characteristic p > 0 and study the essential dimension of inseparable extensions L/K. Here the degree n = [L:K] is replaced by a pair (n, e) which …

Essential dimension and pro-finite group schemes

A. Vistoli observed that, if Grothendieck's section conjecture is true and $X$ is a smooth hyperbolic curve over a field finitely generated over $\mathbb{Q}$, then $\underlineπ_{1}(X)$ should somehow have essential dimension $1$. We prove that an infinite, pro-finite étale group scheme always has infinite essential dimension. We introduce a variant of essential dimension, the fce dimension $\operatorname{fced} G$ of a pro-finite group scheme $G$, which naturally coincides with $\operatorname{ed} G$ if $G$ is finite but has a better behaviour in the pro-finite case. Grothendieck's section conje…

约化群本质维数的下界

Lower bounds on the essential dimension of reductive groups

本文提出一种证明分裂约化群本质维数下界的新技术，加强了包括例外群 E8 在内的多种单代数群的下界，并以简化证明恢复了 PGL_n 的 Merkurjev 下界。

线性群的基本 $2$-维数

The essential $2$-dimension of the linear groups

本文计算了线性群在特定条件下的基本2-维数，涵盖一般线性群、射影一般线性群及特殊线性群等，并给出了精确结果。

A lower bound on the essential dimension of a connected linear group

Let G be a connected linear algebraic group defined over an algebraically closed field k and H be a finite abelian subgroup of G whose order is prime to char(k). We show that the essential dimension of G is bounded from below by rank(H) - rank C_G(H)^0, where rank C_G(H)^0 denotes the rank of the maximal torus in the centralizer C_G(H). This inequality, conjectured by J.-P. Serre, generalizes previous results of Reichstein -- Youssin (where char(k) is assumed to be 0 and C_G(H) to be finite) and Chernousov -- Serre (where H is assumed to be a 2-group).

Essential dimension, spinor groups and quadratic forms

We prove that the essential dimension of the spinor group Spin_n grows exponentially with n; in particular, we give a precise formula for this essential dimension when n is not divisible by 4. We use this result to show that the number of 3-fold Pfister forms needed to represent the Witt class of a general quadratic form of rank n with trivial discriminant and Hasse-Witt invariant grows exponentially with n. 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 "Essen…

On the essential dimension of an algebraic group whose connected component is a torus

Let $p$ be a prime integer, $k$ be a $p$-closed field of characteristic $\neq p$, $T$ be a torus defined over $k$, $F$ be a finite $p$-group, and $1\to T \to G \to F \to 1$ be an exact sequence of algebraic groups. Extending earlier work of N. Karpenko and A. Merkurjev, R. Lötscher, M. MacDonald, A. Meyer, and the first author showed that \[\min\dim(V) - \min\dim(G) \leqslant \text{ed}(G; p) \leqslant \min \dim(W) - \dim(G),\] where $V$ and $W$ range over the $p$-faithful and $p$-generically free $k$-representations of $G$, respectively. They conjectured that the upper bound is, in fact, sharp…

非连通群概形下$\u003cmathcal{G}\u003e$-丛的模空间与本质有限丛的非稠密性

Moduli of $\mathcal{G}$-bundles under nonconnected group schemes and nondensity of essentially finite bundles

该研究证明了光滑射影曲线上非连通约化群概形的主$\u003cmathcal{G}\u003e$-丛存在射影良好模空间，且模栈可分解为有限子栈，进而证明除环面群或亏格小于2的曲线外，0次半稳定$G$-丛模空间中本质有限$G$-丛不稠密。

有限群的本质维数与预解式次数已知界综述

A Summary of Known Bounds on the Essential Dimension and Resolvent Degree of Finite Groups

本文综述特征 0 下有限群本质维数 ed(G) 与预解式次数 RD(G) 的已知界，并在附录中改进 PSL(2,F_11) 的 RD 界。

Remarks on approximate decompositions of the diagonal

In this paper, we investigate, for varieties over $\mathbb C$ with trivial group of $0$-cycles, the gap between essential $\mathrm{CH}_0$-dimension $2$ and essential $\mathrm{CH}_0$-dimension $0$. In particular, we present sufficient (and necessary) conditions for a variety with trivial group of $0$-cycles and essential $\mathrm{CH}_0$-dimension $\leq 2$ to have, in fact, essential $\mathrm{CH}_0$-dimension $0$.

On the essential (p)-dimension of parabolic bundles on a curve

We study the essential dimension and essential p-dimension of the moduli stack of vector bundles over a smooth orbifold curve containing a rational point. We improve the known bounds on this essential dimension and obtain an equality modulo the famous conjecture of Colliot-Thelene, Karpenko and Merkurjev. In the case of essential p-dimension we obtain an equality.

Upper bounds for the essential dimension of the moduli stack of ${\sl_{n}}$-bundles over a curve

We find upper bounds for the essential dimension of various moduli stacks of $\sln$-bundles over a curve. When $n$ is a prime power, our calculation computes the essential dimension of the stack of stable bundles exactly and the essential dimension is not equal to the dimension in this case.

Essential dimension in mixed characteristic

Suppose $G$ is a finite group and $p$ is either a prime number or $0$. For $p$ positive, we say that $G$ is weakly tame at $p$ if $G$ has no non-trivial normal $p$-subgroups. By convention we say that every finite group is weakly tame at $0$. Now suppose that $G$ is a finite group which is weakly tame at the residue characteristic of a discrete valuation ring $R$. Our main result shows that the essential dimension of $G$ over the fraction field $K$ of $R$ is at least as large as the essential dimension of $G$ over the residue field $k$. We also prove a more general statement of this type for a…

Essential dimension of Hermitian spaces

Given an hermitian space we compute its essential dimension, Chow motive and prove its incompressibility in certain dimensions.

Essential dimension of the moduli stack of polarized K3 surfaces

We will calculate the essential dimension of the moduli stack of polarized K3 surfaces, including the positive char case.

有限p-挠子的本质维数与忠实秩

Essential Dimension and Faithful Rank of Finite p-Gerbes

该研究将Karpenko–Merkurjev定理推广至几何惯性群为p-群的有限p-挠子，明确了其本质维数与忠实秩的关系，引入商压缩维数并建立了其与本质p-维数的界。

The Essential Dimension of Stacks of Parabolic Vector Bundles over Curves

We find upper bounds on the essential dimension of the moduli stack of parabolic vector bundles over a curve. When there is no parabolic structure, we improve the known upper bound on the essential dimension of the usual moduli stack. Our calculations also give lower bounds on the essential dimension of the semistable locus inside the moduli stack of vector bundles of rank $r$ and degree $d$ without parabolic structure.

Alexander polynomials: Essential variables and multiplicities

We explore the codimension one strata in the degree-one cohomology jumping loci of a finitely generated group, through the prism of the multivariable Alexander polynomial. As an application, we give new criteria that must be satisfied by fundamental groups of smooth, quasi-projective complex varieties. These criteria establish precisely which fundamental groups of boundary manifolds of complex line arrangements are quasi-projective. We also give sharp upper bounds for the twisted Betti ranks of a group, in terms of multiplicities constructed from the Alexander polynomial. For Seifert links in …

Vector bundles trivialized by proper morphisms and the fundamental group scheme

Let X be a smooth projective variety defined over an algebraically closed field k. Nori constructed a category of vector bundles on X, called essentially finite vector bundles, which is reminiscent of the category of representations of the fundamental group (in characteristic zero). In fact, this category is equivalent to the category of representations of a pro--finite group scheme which controls all finite torsors. We show that essentially finite vector bundles coincide with those which become trivial after being pulled back by some proper and surjective morphism to X.

$B$型半单群的本质维数

Essential dimension of semisimple groups of type $B$

本文确定了特征0域上特定$B$型半单群的本质维数，涵盖了$n_i\geq 7$的一般情况以及部分低维特殊情形。

A Valuation Theoretic Approach to Essential Dimension

In this essay we explore the notion of essential dimension using the theory of valuations of fields. Given a field extension K/k and a valuation on K that is trivial on k, we prove that the rank of the valuation cannot exceed the transcendence degree trdeg K. We use this inequality to prove lower bounds on the essential dimension in some interesting situations. We study orbits of a torus action and find a formula for the essential dimension of the functor of these orbits.

一种新的模形式及其本质变换规律的证明

A new kind of automorphic form and a proof of the essential transformation laws

本文提出了一种新的向量模形式，并证明了其在模群生成元下的函数方程。

关于曲线上辛向量丛的本质维数

On the essential dimension of symplectic vector bundles over curves

该研究针对特征零域上亏格≥2的光滑几何连通射影曲线，借助辛丛模叠的通用gerbe周期为2的特性，精确计算出辛丛模叠的本质维数，弥补了向量丛情形下无法精确计算的相关问题。

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 …

Galois Closure of Essentially Finite Morphisms

Let $X$ be a reduced connected $k$-scheme pointed at a rational point $x \in X(k)$. By using tannakian techniques we construct the Galois closure of an essentially finite $k$-morphism $f:Y\to X$ satisfying the condition $H^0(Y,\mathcal{O}_Y)=k$; this Galois closure is a torsor $p:\hat{X}_Y\to X$ dominating $f$ by an $X$-morphism $λ:\hat{X}_Y\to Y$ and universal for this property. Moreover we show that $λ:\hat{X}_Y\to Y$ is a torsor under some finite group scheme we describe. Furthermore we prove that the direct image of an essentially finite vector bundle over $Y$ is still an essentially finit…

本质维度在专门化下的行为

The behavior of essential dimension under specialization

该研究证明了在特定条件下，覆盖的闭点本质维度不超过通用点的本质维度，并应用于混合特征和等特征场景。

Essential dimension and genericity for quiver representations

We study the essential dimension of representations of a fixed quiver with given dimension vector. We also consider the question of when the genericity property holds, i.e., when essential dimension and generic essential dimension agree. We classify the quivers satisfying the genericity property for every dimension vector and show that for every wild quiver the genericity property holds for infinitely many of its Schur roots. We also construct a large class of examples, where the genericity property fails. Our results are particularly detailed in the case of Kronecker quivers.

本质有限 $G$-torsors

Essentially finite $G$-torsors

本文将本质有限向量丛推广至任意连通约化群的 $G$-torsors，给出其 Tannakian 解释，并研究了度为 0 的本质有限 $G$-torsors 在半稳定 torsors 空间中的稠密性。

The period-index problem for elliptic curves and the essential dimension of Picard stacks

In this paper, we will consider the period index problems of elliptic curves and introduce a value called generic index which is closed related to the essential dimension of Picard stacks. In particular, we will use examples to see that usually the period and generic index are different. Our construction is different with Cassel's example.

本质p-维数与陈数

Essential p-dimension and Chern numbers

本文研究本质p-维数与陈数，证明有限p群在特定光滑射影簇上的作用可分解，推导对基域算术敏感的不动点定理，得到Cremona群循环p子群阶的界，关键利用Karpenko与Merkurjev的p群本质p-维数计算。

同源映射的本质维数

Essential dimension of isogenies

本文研究复阿贝尔簇同源映射的本质维数，给出在许多情形下精确的下界，并证明对所有 m≥2 的乘以 m 映射均不可压缩，从而证实 Brosnan 猜想。

On the essential dimension of coherent sheaves

We characterize all fields of definition for a given coherent sheaf over a projective scheme in terms of projective modules over a finite-dimensional endomorphism algebra. This yields general results on the essential dimension of such sheaves. Applying them to vector bundles over a smooth projective curve C, we obtain an upper bound for the essential dimension of their moduli stack. The upper bound is sharp if the conjecture of Colliot-Thélène, Karpenko and Merkurjev holds. We find that the genericity property proved for Deligne-Mumford stacks by Brosnan, Reichstein and Vistoli still holds for…

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 stac…