Najmuddin Fakhruddin
We prove that if a finite group scheme $G$ over a field $k$ has essential dimension one, then it embeds in $PGL_{2/k}$. We use this to give an explicit classification of all infinitesimal group schemes of essential dimension one over any field and a characterisation of all finite group schemes of essential dimension one over algebraically closed fields.
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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 $\geq 2$.
Strict homotopy invariance via compactified homotopies and correspondences, and fibres of essentially smooth schemes over one-dimensional base schemes
本研究发展一维基概形上的紧化对应与同伦技术，阐明Voevodsky严格同伦不变性定理等的特征假设性质，还得到广义动机上同伦Cousin复形相关猜想的类似结论及一类无环性结果。
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.
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.
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.
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。
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…
Let f: X -> Z be a separated essentially-finite-type flat map of noetherian schemes, and δ: X --> X \times_Z X the diagonal map. The fundamental class C_f (globalizing residues) is a map from the relative Hochschild functor Lδ^*δ_* f^* to the relative dualizing functor f^! A compatibility between this C_f and derived tensor product is shown. The main result is that, in a suitable sense, C_f generalizes Verdier's classical isomorphism for smooth f with fibers of dimension d, an isomorphism that binds f^! to relative d-forms.
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.
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.
Fix a noetherian scheme S. For any flat map f: X->Y of separated essentially-finite-type perfect S-schemes we define a canonical derived-category map c(f):\H(X)->f^!\H(Y), the fundamental class of f, where \H(Z) is the (pre-)Hochschild complex of an S-scheme Z and f^! is the twisted inverse image coming from Grothendieck duality theory. When Y=S and f is essentially smooth of relative dimension n, this gives an isomorphism from n-th degree relative differential forms [ =H^{-n}(\H(X)) ] to f^!O_S[-n]. The basic results concern transitivity of c(-) vis-à-vis compositions X->Y->Z, and compatibili…
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…
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.
We classify quasi-simple finite groups of essential dimension 3.
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.
Given an hermitian space we compute its essential dimension, Chow motive and prove its incompressibility in certain dimensions.
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.
We will calculate the essential dimension of the moduli stack of polarized K3 surfaces, including the positive char case.
We classify all finite groups of essential dimension 2 over an algebraically closed field of characteristic 0.
We provide a coherent overview of a number of recent results obtained by the authors in the theory of schemes defined over the field with one element. Essentially, this theory encompasses the study of a functor which maps certain geometries including graphs to Deitmar schemes with additional structure, as such introducing a new zeta function for graphs. The functor is then used to determine automorphism groups of the Deitmar schemes and base extensions to fields.
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.
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.
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.
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.
In this note, we prove that the F-fundamental group scheme is birational invariant for smooth projective varieties. We prove that the F-fundamental group scheme is naturally a quotient of the Nori fundamental group scheme. For elliptic curves, it turns out that the F-fundamental group scheme and the Nori fundamental group scheme coincides. We also consider an extension of the Nori fundamental group scheme in positive characteristic using semi-essentially finite vector bundles and prove that in this way, we do not get a non-trivial extension of the Nori fundamental group scheme for elliptic cur…
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.
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。
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。
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.
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.
Let $f:X\to Y$ be a surjective morphism of integral schemes. Then $X$ is said to be quasi-galois closed over $Y$ by $f$ if $X$ has a unique conjugate over $Y$ in an algebraically closed field. Such a notion has been applied to the computation of étale fundamental groups. In this paper we will use affine coverings with values in a fixed field to discuss quasi-galois closed and then give a sufficient and essential condition for quasi-galois closed. Here, we will avoid using affine structures on a scheme since their definition looks copious and fussy.
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。
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…
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…
A ringed finite space is a ringed space whose underlying topological space is finite. The category of ringed finite spaces contains, fully faithfully, the category of finite topological spaces and the category of affine schemes. Any ringed space, endowed with a finite open covering, produces a ringed finite space. We introduce the notions of schematic finite space and schematic morphism, showing that they behave, with respect to quasi-coherence, like schemes and morphisms of schemes do. Finally, we construct a fully faithful and essentially surjective functor from a localization of a full subc…
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 fundamental group schemes of certain quotient varieties
本文在$F$-divided和本质有限基本群概形的设定下，证明了Armstrong关于商空间基本群定理的类似结果。
Galois self-covers of projective spaces and essential dimensions
分类射影空间的伽罗瓦自覆盖，并证明每个非平凡自覆盖的本质维数达到最大值n。
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.
We use a recent advance in birational geometry to prove new lower bounds on the essential dimension of some finite groups.
We study functors underlying derived Hochschild cohomology, also called Shukla cohomology, of a commutative algebra S essentially of finite type and of finite flat dimension over a commutative noetherian ring K. We construct a complex of S-modules D, and natural reduction isomorphisms Ext^*_{S\otimes^L_{K}S}(S|K;M\otimes^L_{K}N) ~ Ext^*_S(RHom_S(M,D),N) for all complexes of S-modules N and all complexes M of finite flat dimension over K whose homology H(M) is finitely generated over S; such isomorphisms determine D up to derived isomorphism. Using Grothendieck duality theory we establish analo…
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.
Let K be an arbitrary field. We will determine explicitly all the nontrivial finite groups of essential dimension one over K.
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.
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.
We study the essential dimension of a finite group G over a field K. A generalization of the central extension theorem of Buhler and Reichstein (Compositio Math. 106 (1997) 159-179, Theorem 5.3) is obtained. We also get lower bounds of essential dimension of symmetric groups and alternating groups over field of any characteristic.
We prove a new upper bound on the essential p-dimension of the projective linear group PGLn.
We prove the existence of a Galois closure for towers of torsors under finite group schemes over a proper, geometrically connected and geometrically reduced algebraic stack $X$ over a field $k$. This is done by describing the Nori fundamental gerbe of an essentially finite cover of $X$. A similar result is also obtained for the $S$-fundamental gerbe.
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…
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。
By a result of Biswas and Dos Santos, on a smooth and projective variety over an algebraically closed field, a vector bundle trivialized by a proper and surjective map is essentially finite, that is it corresponds to a representation of the Nori fundamental group scheme. In this paper we obtain similar results for non-proper non-smooth algebraic stacks over arbitrary fields of characteristic $p>0$. As by-product we have the following partial generalization of the Biswas-Dos Santos' result in positive characteristic: on a pseudo-proper and inflexible stack of finite type over $k$ a vector bundl…
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…
We prove universality theorems ("Murphy's Laws") for representation schemes of fundamental groups of closed 3-dimensional manifolds. We show that germs of SL(2,C)-representation schemes of such groups are essentially the same as germs of schemes of over rational numbers.
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。
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。
Hom schemes for algebraic groups
本文拓展了SGA3中约化群同态函子可表的经典结论，覆盖G非约化的情形且证明更简洁，结论适用于任意基抛物子群，还关联了Serre完全可约性理论、解答Furter-Kraft问题并给出示例。
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 …
Springer isomorphisms over a general base scheme
本文证明了一般基概形上约化群概形存在Springer同构，通过研究纤维正则截面的中心化子并证明其平坦性，并论证了条件的本质最优性。
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。
The behavior of essential dimension under specialization II
该研究针对域上线性代数群的本原一般自由G-簇族，证明温和假设下几何纤维本质维数仅在基簇可数扎里斯基闭子集并上下降、其余区域恒定，并给出相关应用。
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