### Citations

96 |
Vorlesungen uber das Ikosaeder und die Auflosung der Gleichungen vom fumften Grade,
- Klein
- 1884
(Show Context)
Citation Context ...due to A. Merkurjev; see [BF03]. In special cases this notion was investigated much earlier. To the best of my knowledge, the first non-trivial result related to essential dimension is dueto F. Klein =-=[Kl1884]-=-. Inourterminology, Klein showed that the essential dimension of the symmetric group S5 over k = C, is 2. (Klein referred to this result as “Kronecker’s theorem”, so it may in fact go back even furthe... |

78 | On the notion of essential dimension for algebraic groups, - Reichstein - 2000 |

73 | Iskovskikh, “Finite subgroups of the plane Cremona group” math/0610595
- Dolgachev, A
(Show Context)
Citation Context ...mplicated than Klein’s classification of rational curves but one can use it to determine, at least in principle, which finite groups G can act on a rational surface and describe all such actions; cf. =-=[DI06]-=-. Once all minimal rational G-surfaces X are accounted for, one then needs to decide, for each X, whether or not the G-action is versal, i.e., whether or not a dominant rational G-equivariant map (6.1... |

72 | On the essential dimension of a finite group,
- Buhler, Reichstein
- 1997
(Show Context)
Citation Context ... algebraic object is the minimal number of independent parameters one needs to define it. This notion was initially introduced in the context where the objects in question are finite field extensions =-=[BuR97]-=-. Essential dimension has since been investigated in several broader contexts, by a range of techniques, and has been found to have interesting and surprising connections to many problems in algebra a... |

64 |
Essential dimension: a functorial point of view (after
- Berhuy, Favi
- 2003
(Show Context)
Citation Context ... dimension was originally introduced in this context (and only in characteristic 0); see[BuR97,Rei00,RY00]. Theabovedefinitionofessential dimension for a general functor F is due to A. Merkurjev; see =-=[BF03]-=-. In special cases this notion was investigated much earlier. To the best of my knowledge, the first non-trivial result related to essential dimension is dueto F. Klein [Kl1884]. Inourterminology, Kle... |

63 | Essential dimensions of algebraic groups and a resolution theorem for G-varieties (with an appendix by Janos Kollar and Endre Szabo),
- Reichstein, Youssin
- 2000
(Show Context)
Citation Context ...ial dimension of the Brauer class of A is also 2s. 3. The fixed point method The following lower bound on ed(G) was conjectured by Serre and proved in[GR07]. Earlierversionsofthistheoremhaveappearedin=-=[RY00]-=-and[CS06]. Theorem 3.1. If G is connected, A is a finite abelian subgroup of G and char(k) does not divide |A|, then edk(G) ≥ rank(A)−rank C 0 G (A).ESSENTIAL DIMENSION 7 Here rank(A) stands for the ... |

57 | Galois Cohomology, Springer Monographs in Mathematics, - Serre - 2002 |

53 |
Torsion homologique et sections rationnelles, exposé 5 in Anneaux de Chow et applications
- Grothendieck
(Show Context)
Citation Context ...an algebraic group G over k is called special if H 1 (K,G) = 0 for every field extension K/k. Over an algebraically closed field of characteristic zero these groups were classified by A. Grothendieck =-=[Gro58]-=- in the 1950s. The problem of computing the essential dimension of an algebraic group may be viewed as a natural extension of the problem of classifying special groups. 2. First examples Recall that a... |

53 | Canonical pdimension of algebraic groups
- Karpenko, Merkurjev
(Show Context)
Citation Context ...G and is denoted by cdim(G). Clearly 0 ≤ cdim(G) ≤ dim(G) and cdim(G) = 0 if and only if G is special. For a detailed discussion of the notion of canonical dimension, we refer the reader to [BerR05], =-=[KM06]-=- and [Me09].16 ZINOVY REICHSTEIN Computing the canonical dimension cdim(G) of an algebraic group G is a largely open Type 2 problem. The associated Type 1 problem of computing the canonical p-dimensi... |

47 |
Cohomologie galoisienne: progrès et problèmes, Astérisque
- Serre
- 1995
(Show Context)
Citation Context ... the least common multiple of nα taken over all K/k and all α ∈ H1 (K,G). One can show that nG = nαver, where αver ∈ H1 (Kversal,G) is a versal G-torsor. One can also show, using a theorem of J. Tits =-=[Se95]-=-, that the prime divisors of nG are precisely the exceptional primes of G. TheproblemofcomputingnG andmoregenerally, of nα forα ∈ H1 (K,G) can thus be rephrased as follows. Given an exceptional prime ... |

44 |
Torsion in reductive groups
- STEINBERG
- 1975
(Show Context)
Citation Context ...integer. A closer look reveals that in every single case the argument can be modified to show that ed(G;p) ≥ d, for a suitable prime p. (Usually p is a so-called “exceptional prime” for G; see, e.g., =-=[St75]-=- or [Me09]. Sometimes there is more than one such prime.) In particular, the arguments we used in Examples 2.4, 2.5, 2.6 and 2.8 show that ed(G2;2) = 3, ed(On;2) = n, ed(µ r p;p) = r and ed(PGLps;p) ≥... |

41 | The behaviour at infinity of the Bruhat decomposition - Brion - 1998 |

40 | On anisotropy of orthogonal involutions. - Karpenko - 2000 |

38 | Essential dimension of finite p-groups
- Karpenko, Merkurjev
(Show Context)
Citation Context ...ounds. Let (4.1) 1 → C → G → G → 1 be an exact sequence of algebraic groups over k such that C is central in G and isomorphic to µ r p for some r ≥ 1. Given a character χ: C → µ p, we will, following =-=[KM07]-=-, denote by Rep χ the set of irreducible representations φ: G → GL(V), defined over k, such that φ(c) = χ(c)IdV for every c ∈ C. Theorem 4.1. Assume that k is a field of characteristic ̸= p containing... |

31 |
On central division algebras
- Amitsur
- 1972
(Show Context)
Citation Context ...ry “homogenization” argument I discussed in the previous section has not (yet?) yielded a lower bound on ed(G;p).ESSENTIAL DIMENSION 15 this one by the primary decomposition theorem. In 1972 Amitsur =-=[Am72]-=- showed that for r ≥ 3 a generic division algebra U(p r ) of degree p r is not a crossed product, solving a long-standing open problem. L. H. Rowen and D. J. Saltman [RS92, Theorem 2.2] modified Amits... |

30 | On the notion of canonical dimension for algebraic groups
- Berhuy, Reichstein
(Show Context)
Citation Context ...ension of G and is denoted by cdim(G). Clearly 0 ≤ cdim(G) ≤ dim(G) and cdim(G) = 0 if and only if G is special. For a detailed discussion of the notion of canonical dimension, we refer the reader to =-=[BerR05]-=-, [KM06] and [Me09].16 ZINOVY REICHSTEIN Computing the canonical dimension cdim(G) of an algebraic group G is a largely open Type 2 problem. The associated Type 1 problem of computing the canonical p... |

28 | Simple finite subgroups of the Cremona group of rank 3
- Prokhorov
(Show Context)
Citation Context ...e Enriques-Manin-Iskovskikh classification of rational surfaces in higher dimensions. Nevertheless, in dimension 3 one can sometimes use Mori theory to get a handle on X. In particular, Yu. Prokhorov =-=[Pr09]-=- recently classified the finite simple groups with faithful actions on rationally connected threefolds. This classification was used by Duncan [Du09b] to prove the following theorem, which is out of t... |

26 |
Cohomological invariants: exceptional groups and spin groups, vol. 200, Memoirs
- Garibaldi
- 2009
(Show Context)
Citation Context ...n(n−1) 2 +n, if n ≡ 0 (mod 4). Here 2m is the largest power of 2 dividing n. We remark that M. Rost and S. Garibaldi have computed the essential dimension of Spin n for every n ≤ 14; see [Rost06] and =-=[Gar09]-=-. Proof outline. The lower bounds (e.g., ed(Spin n) ≥ 2 (n−1)/2 − n(n−1) 2 , in part(a)) arevalid whenever char(k) ̸= 2; they can bededucedeither directly fromTheorem4.1orbyapplyingtheinequality (2.5)... |

25 | Structurable algebras and groups of type E6 and E7. - Garibaldi - 2001 |

25 | Le groupe de Cremona et ses sous-groupes finis. Séminaire Bourbaki. Volume 2008/2009 - Serre |

24 | Resolving G-torsors by abelian base extensions
- Chernousov, Gille, et al.
(Show Context)
Citation Context ...ed as in the previous paragraph: we will use Theorem 3.2 to find an A-fixed point on a smooth complete model of Z, then use Lemma 3.3 to show that dim(Z) ≥ rank(A). Let me now fill in the details. By =-=[CGR06]-=- Y is birationally isomorphic to G× S Z, where S is a finite subgroup of G and Z is an algebraic variety equipped with a faithful S-action. (A priori Z does not carry an A-action; however, we will sho... |

23 | Lower bounds for essential dimensions via orthogonal representations
- Chernousov, Serre
(Show Context)
Citation Context ...sion of the Brauer class of A is also 2s. 3. The fixed point method The following lower bound on ed(G) was conjectured by Serre and proved in[GR07]. Earlierversionsofthistheoremhaveappearedin[RY00]and=-=[CS06]-=-. Theorem 3.1. If G is connected, A is a finite abelian subgroup of G and char(k) does not divide |A|, then edk(G) ≥ rank(A)−rank C 0 G (A).ESSENTIAL DIMENSION 7 Here rank(A) stands for the minimal n... |

19 |
Essential dimension of algebraic groups and integral representations of Weyl groups’,
- Lemire
- 2004
(Show Context)
Citation Context ... on edC(E8;2) is 9 (see Corollary 3.6(i)) but the best upper bound I know is edC(E8;2) ≤ 120. The essential dimension ed(G) for these groups is largely uncharted territory, beyond the upper bounds in =-=[Lem04]-=-. 7.9. Groups whose connected component is a torus. Let G be an algebraic group over k and p be a prime. We say that a linear representation φ: G → GL(V) is p-faithful (respectively, p-generically fre... |

18 |
Espaces fibrés algébriques, Anneaux de Chow et applications
- Serre
- 1942
(Show Context)
Citation Context ...jective linear group PGLn remain largely open; see Section 7. If k is an algebraically closed field then groups of essential dimension zero are precisely the special groups, introduced by J.-P. Serre =-=[Se58]-=-. Recall that an algebraic group G over k is called special if H 1 (K,G) = 0 for every field extension K/k. Over an algebraically closed field of characteristic zero these groups were classified by A.... |

17 | The essential dimension of the normalizer of a maximal torus in the projective linear group’, - Meyer, Reichstein - 2009 |

17 | les degrés des extensions de corps déployant les groupes algébriques simples - Sur - 1992 |

15 | Essential dimension, spinor groups, and quadratic forms
- Brosnan, Reichstein, et al.
(Show Context)
Citation Context ...int here is to check that these representations are generically free. In characteristic 0 this isduetoE.AndreevandV. Popov [AP71] forn ≥ 29andtoA.Popov [Po85] in the remaining cases. For details, see =-=[BRV10a]-=- and (for the lower bound in part (c)) [Me09, Theorem 4.9]. □ To convey the flavor of the proof of Theorem 4.1, I will consider a special case, where G is finite and r = 1. That is, I will start with ... |

15 |
On the essential dimension of cyclic p-groups
- Florence
(Show Context)
Citation Context ...Recall that H 2 (K,µ p) is naturally isomorphic to the p-torsion subgroup of the Brauer group Br(K), so that it makes sense to talk about the index. I will now outline an argument, due to M. Florence =-=[Fl07]-=-, which deduces the inequality (4.5) from these two theorems. To begin with, let us choose a faithful representation V of G, where C acts by scalar multiplication. In particular, we can induce V from ... |

15 | Fields of definition for division algebras - Lorenz, Reichstein, et al. |

14 | Essential dimension of moduli of curves and other algebraic stacks, with an appendix by Najmuddin Fakhruddin
- Brosnan, Reichstein, et al.
- 2009
(Show Context)
Citation Context ...a completely reducible faithful k-representation. The only known proof of Theorem 4.1 in full generality uses the notion of essential dimension for an algebraic stack, introduced in [BRV07]; cf. also =-=[BRV10b]-=-. For details, see [Me09, Theorem 4.8 and Example 3.7], in combination with [KM07, Theorem 4.4 and Remark 4.5]. 5. Essential dimension at p and two types of problems Let p be a prime integer. I will s... |

14 | Prime to p extensions of division algebra’, - Rowen, Saltman - 1992 |

13 |
On a theorem of Hermite and
- Reichstein
- 1999
(Show Context)
Citation Context ...ver k, K = k(a1,...,a2s), and A = (a1,a2)p ⊗K ⊗···⊗(a2s−1,a2s)p is a tensor product of s symbol algebras of degree p then one can write out φ(A) explicitly and show that it is anisotropic over K; see =-=[Rei99]-=-. Lemma 2.2 and Example 1.2 now tell us that edk(A) ≥ ed k (A) ≥ 2s. Since trdeg k(K) = 2s, we conclude that, in fact (2.6) edk(A) = 2s and consequently, edk(PGLps) ≥ 2s. The following alternative app... |

12 | Comparison of some field invariants - Kahn |

12 | Cohomological invariants of odd degree Jordan algebras - MacDonald |

10 | Essential dimensions of A7 and S7
- Duncan
- 2010
(Show Context)
Citation Context ...to get a handle on X. In particular, Yu. Prokhorov [Pr09] recently classified the finite simple groups with faithful actions on rationally connected threefolds. This classification was used by Duncan =-=[Du09b]-=- to prove the following theorem, which is out of the reach of all previously existing methods. Theorem 6.4. Let k be a field of characteristic 0. Then edk(A7) = edk(S7) = 4. Note that ed(A7;p) ≤ ed(S7... |

10 | A lower bound on the essential dimension of a connected linear group
- Gille, Reichstein
(Show Context)
Citation Context ... of this approach is that it shows that the essential dimension of the Brauer class of A is also 2s. 3. The fixed point method The following lower bound on ed(G) was conjectured by Serre and proved in=-=[GR07]-=-. Earlierversionsofthistheoremhaveappearedin[RY00]and[CS06]. Theorem 3.1. If G is connected, A is a finite abelian subgroup of G and char(k) does not divide |A|, then edk(G) ≥ rank(A)−rank C 0 G (A).... |

10 | On the essential dimension of some semi-direct products, Canad
- Ledet
(Show Context)
Citation Context ...e of computing edk(Z/nZ;p) we are allowed to replace k by k(ζp); then formula (7.3) applies. If we do not assume that ζp ∈ k then the best currently known upper bound on edk(Z/p r Z), due to A. Ledet =-=[Led02]-=-, is edk(Z/p r Z) ≤ ϕ(d)p e . Here [k(ζp r) : k] = dpe , where d divides p−1, and ϕ is the Euler ϕ-function. Nowletussupposechar(k) = p > 0. Hereitiseasytoseethatedk(Z/p r Z) ≤ r; Ledet [Led04] conjec... |

9 |
Essential dimension, in Quadratic forms
- Merkurjev
- 2009
(Show Context)
Citation Context ... closer look reveals that in every single case the argument can be modified to show that ed(G;p) ≥ d, for a suitable prime p. (Usually p is a so-called “exceptional prime” for G; see, e.g., [St75] or =-=[Me09]-=-. Sometimes there is more than one such prime.) In particular, the arguments we used in Examples 2.4, 2.5, 2.6 and 2.8 show that ed(G2;2) = 3, ed(On;2) = n, ed(µ r p;p) = r and ed(PGLps;p) ≥ 2s, respe... |

9 | Canonical p-dimensions of algebraic groups and degrees of basic polynomial invariants
- Zainoulline
(Show Context)
Citation Context ...of an algebraic group G is a largely open Type 2 problem. The associated Type 1 problem of computing the canonical p-dimension cdim(G;p) has been solved by KarpenkoMerkurjev [KM06] and K. Zainoulline =-=[Zai07]-=-. 6. Finite groups of low essential dimension Suppose we would like to determine the essential dimension of a finite group G. To keep things simple, we will assume throughout this section that, unless... |

8 |
Non-commutative affine rings, Atti
- Procesi
- 1967
(Show Context)
Citation Context ...dim(PGLn) is only known if n = 6 or a prime power. 7.6. Essential dimension of PGLn. Problem. What is ed(PGLn;p)? ed(PGLn)? As I mentioned in Section 1, this problem originated in the work of Procesi =-=[Pr67]-=-; for a more detailed history, see [MR09a, MR09b]. The second question appears to be out of reach at the moment, except for a few small values of n. However, there has been a great deal of progress on... |

8 | The torsion index of E8 and other groups - Totaro |

7 |
Maximal indexes of Tits algebras
- Merkurjev
- 1996
(Show Context)
Citation Context ...Severi variety of prime power index p m , over a field K and let f: X ��� X be a rational map defined over K. Then dimK Im(f) ≥ p m −1. Theorem 4.5. (Merkurjev’sIndexTheorem[KM07,Theorem4.4]; cf. also=-=[Me96]-=-) Let K/k be a field extension, and ∂K: H 1 (K,G) → H 2 (K,µ p) be the connecting map induced by the short exact sequence (4.4). Then the maximal value of the index of ∂K(a), as K ranges over all fiel... |

7 | An upper bound on the essential dimension of a central simple algebra’,
- Meyer, Reichstein
- 2011
(Show Context)
Citation Context ...e have (r −1)p r +1 ≤ ed(PGLpr;p) ≤ p2r−2 +1. ThelowerboundisduetoMerkurjev[Me10b]; theupperboundisprovedina recentpreprintofA.Ruozzi[Ru10]. (Aweakerupperbound,ed(PGLn;p) ≤ 2p2r−2−pr +1, is proved in =-=[MR09b]-=-.) In particular, ed(PGLp2;p) = p2 +1; see [Me10a]. Note that the argument in [Me10b] shows that if A is a generic (Z/pZ) rcrossed product then ed(A;p) = (r−1)pr +1. As mentioned in Example 5.2, forr ... |

6 | Essential dimension and algebraic stacks
- Brosnan, Reichstein, et al.
(Show Context)
Citation Context ...at it should have a completely reducible faithful k-representation. The only known proof of Theorem 4.1 in full generality uses the notion of essential dimension for an algebraic stack, introduced in =-=[BRV07]-=-; cf. also [BRV10b]. For details, see [Me09, Theorem 4.8 and Example 3.7], in combination with [KM07, Theorem 4.4 and Remark 4.5]. 5. Essential dimension at p and two types of problems Let p be a prim... |

6 | Essential p-dimension of PGL(p2)’,
- Merkurjev
- 2010
(Show Context)
Citation Context ...owerboundisduetoMerkurjev[Me10b]; theupperboundisprovedina recentpreprintofA.Ruozzi[Ru10]. (Aweakerupperbound,ed(PGLn;p) ≤ 2p2r−2−pr +1, is proved in [MR09b].) In particular, ed(PGLp2;p) = p2 +1; see =-=[Me10a]-=-. Note that the argument in [Me10b] shows that if A is a generic (Z/pZ) rcrossed product then ed(A;p) = (r−1)pr +1. As mentioned in Example 5.2, forr ≥ 3ageneral divisionalgebraA/K ofdegreep r isnotac... |

6 | On the essential dimension of infinitesimal group schemes
- Tossici, Vistoli
- 2010
(Show Context)
Citation Context ...rally, essential dimension of finite (but not necessarily smooth) group schemes over a field k of prime characteristic is poorly understood; some interesting results in this direction can be found in =-=[TV10]-=-. 7.4. Quadratic forms. Let us assume that char(k) ̸= 2. The following question is due to J.-P. Serre (private communication, April 2003). Problem. If q is a quadratic form over K/k, is it true that e... |

5 |
The stationary subgroups of points in general position in a representation space of a semisimple Lie group (Russian), Funkcional. Anal. i Prilozen
- Andreev, Popov
- 1971
(Show Context)
Citation Context ...sentation of SOn, viewed as a representation of Spin n via π. The delicate point here is to check that these representations are generically free. In characteristic 0 this isduetoE.AndreevandV. Popov =-=[AP71]-=- forn ≥ 29andtoA.Popov [Po85] in the remaining cases. For details, see [BRV10a] and (for the lower bound in part (c)) [Me09, Theorem 4.9]. □ To convey the flavor of the proof of Theorem 4.1, I will co... |

5 |
On the essential dimension of p-groups, Galois Theory and Modular Forms
- Ledet
(Show Context)
Citation Context ... Ledet [Led02], is edk(Z/p r Z) ≤ ϕ(d)p e . Here [k(ζp r) : k] = dpe , where d divides p−1, and ϕ is the Euler ϕ-function. Nowletussupposechar(k) = p > 0. Hereitiseasytoseethatedk(Z/p r Z) ≤ r; Ledet =-=[Led04]-=- conjectured that equality holds. This seems to be out of reach at the moment, at least for r ≥ 5. More generally, essential dimension of finite (but not necessarily smooth) group schemes over a field... |

5 |
Spinor groups and algebraic coding theory
- Wood
- 1989
(Show Context)
Citation Context ...irst proved by V. Chernousov and J.-P. Serre [CS06], by a different method. I later noticed that it can be deduced from Theorem 3.1 as well; the finite abelian subgroups one uses here can be found in =-=[Woo89]-=-. Remark 3.7. The inequalities in parts (a), (b), (d), (e) and (f) can be recovered by applying Lemma 2.3 to suitable cohomological invariants. For parts (b) and (d), this is done in Examples 2.8 and ... |

4 |
Merkurjev: Rational surfaces and the canonical dimension of the group PGL6. Algebra i Analiz 19
- Colliot-Thélène, Karpenko, et al.
- 2007
(Show Context)
Citation Context ...dimension cdim(X) is the minimal dimension of the image of a K-rational map X ��� X. For a detailed discussion of this notion, see [KM06] and [Me09]. Conjecture. (Colliot-Thélène, Karpenko, Merkurjev =-=[CKM08]-=-) Suppose X is a Brauer-Severi variety of index n. If n = p e1 1 ...per r is the prime decomposition of n then cdim(X) = p e1 1 +···+per r −r.ESSENTIAL DIMENSION 21 This is a Type 2 problem. The asso... |

4 |
On the essential dimension and Serre’s conjecture II for exceptional groups
- Kordonskiĭ
(Show Context)
Citation Context ...2.4. For G = F4, the Type 1 problem has been completely solved: ed(F4;2) = 5 (see [MacD08, Section 5]), ed(F4;3) = 3 (see [GR07, Example 9.3]), and ed(F4;p) = 0 for all other primes. It is claimed in =-=[Ko00]-=- that ed(F4) = 5. However, the argument there appears to be incomplete, so the (Type 2) problem of computing ed(F4) remains open. The situation is similar for the simply connected group Esc 6 . The Ty... |

4 | Compression of finite group actions and covariant dimension - Kraft, Schwarz |

4 | Essential p-dimension of algebraic tori
- Lötscher, MacDonald, et al.
- 2009
(Show Context)
Citation Context ... the sake of computing ed(G;p) we may assume that G/G 0 is a pgroup and k is p-closed, i.e., the degree of every finite field extension k ′ /k is a power of p; see [LMMR09, Lemma 3.3]. It is shown in =-=[LMMR09]-=- that (7.4) mindimν −dim(G) ≤ ed(G;p) ≤ mindimρ−dimG, where the two minima are taken over all p-faithful representations ν, and p-generically free representations ρ, respectively. In the case where G ... |

4 |
A lower bound on the essential dimension of simple algebras
- Merkurjev
(Show Context)
Citation Context ...lgebra A of degree p becomes cyclic after a prime-to-p extension. Hence, ed(PGLp;p) = 2; cf. [RY00, Lemma 8.5.7]. For r ≥ 2 we have (r −1)p r +1 ≤ ed(PGLpr;p) ≤ p2r−2 +1. ThelowerboundisduetoMerkurjev=-=[Me10b]-=-; theupperboundisprovedina recentpreprintofA.Ruozzi[Ru10]. (Aweakerupperbound,ed(PGLn;p) ≤ 2p2r−2−pr +1, is proved in [MR09b].) In particular, ed(PGLp2;p) = p2 +1; see [Me10a]. Note that the argument ... |

4 |
Forms with Applications to Geometry and Topology, Cambidge Univ.
- Pfister
- 1995
(Show Context)
Citation Context ...me that k is algebraically closed. Let Formsn,d(K) be the set of homogeneous polynomials of degree d in n variables. If α ∈ Formsn,d(K) is anisotropic over K then by the Tsen-Lang theorem (see, e.g., =-=[Pf95]-=-), n ≤ d ed(α) or equivalently, ed(α) ≥ log d(n). OfparticularinteresttouswillbethefunctorsFG givenbyK → H 1 (K,G), where G is an algebraic group over k. Here, as usual, H 1 (K,G) denotes the set of i... |

4 | Computation of some essential dimensions
- Rost
- 2000
(Show Context)
Citation Context ...alence, has a negative answer. Indeed, assume k contains a primitive 4th root of unity and D = UDk(4) is a universal division algebra of degree 4. Then ed(D) = 5 (see [Me10a, Corollary 1.2], cf. also =-=[Rost00]-=-) while ed M2(D) = 4 (see [LRRS03, Corollary 1.4]). 7.5. Canonical dimension of Brauer-Severi varieties. Let X be a smooth complete variety defined over a field K/k. The canonical dimension cdim(X) is... |

3 |
Application of multihomogeneous covariants to the essential dimension of finite groups. arXiv:0811.3852v1 [math.AG
- Lötscher
- 2008
(Show Context)
Citation Context ...edk(G)−1 ≥ dimK(Im( T f)) ≥ ind( T P(V))−1 = gcd ρ∈Rep ′ dim(ρ)−1, and (4.5) follows. □ Remark 4.6. Now suppose G is finite but r ≥ 1 is arbitrary. The above argument has been modified by R. Lötscher =-=[Lö08]-=- to prove Theorem 4.1 in this more general setting. The proof relies on Theorem 4.5 and a generalization of Theorem 4.4 to the case where X is the direct product of Brauer-Severi varieties X1 ×···×Xr,... |

3 |
On the Galois cohomology of Spin(14), http://www.mathematik.uni-bielefeld.de/˜rost/spin-14.html
- Rost
- 1999
(Show Context)
Citation Context ... 2 (n−2)/2 − n(n−1) 2 +n, if n ≡ 0 (mod 4). Here 2m is the largest power of 2 dividing n. We remark that M. Rost and S. Garibaldi have computed the essential dimension of Spin n for every n ≤ 14; see =-=[Rost06]-=- and [Gar09]. Proof outline. The lower bounds (e.g., ed(Spin n) ≥ 2 (n−1)/2 − n(n−1) 2 , in part(a)) arevalid whenever char(k) ̸= 2; they can bededucedeither directly fromTheorem4.1orbyapplyingtheineq... |

2 |
Finite groups of essential dimension
- Duncan
(Show Context)
Citation Context ...smooth). Such surfaces (called minimal rational G-surfaces) have been classified by Yu. Manin and V. Iskovskikh, following up on classical work of F. Enriques; for details and further references, see =-=[Du09a]-=-. This classification is significantly more complicated than Klein’s classification of rational curves but one can use it to determine, at least in principle, which finite groups G can act on a ration... |

2 |
Finite stationary subgroups in general position of simple linear Lie groups (Russian), Trudy Moskov
- Popov
- 1985
(Show Context)
Citation Context ... representation of Spin n via π. The delicate point here is to check that these representations are generically free. In characteristic 0 this isduetoE.AndreevandV. Popov [AP71] forn ≥ 29andtoA.Popov =-=[Po85]-=- in the remaining cases. For details, see [BRV10a] and (for the lower bound in part (c)) [Me09, Theorem 4.9]. □ To convey the flavor of the proof of Theorem 4.1, I will consider a special case, where ... |

2 |
Compressions of group actions, Invariant theory in all characteristics
- Reichstein
- 2004
(Show Context)
Citation Context ...o any proper subfield of K. Examples of strongly incompressible elements in the case where G is a finite group, K is the function field of an algebraic curve Y over k, and F = H 1 (∗,G), are given in =-=[Rei04]-=-. In these examples α is represented by a (possibly ramified) G-Galois cover X → Y. I do not know any such examples in higher dimensions. Problem. Does there exist a finitely generated field extension... |

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Essential p-dimension of PGL, http://www.mathematik
- Ruozzi
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Citation Context ...ension. Hence, ed(PGLp;p) = 2; cf. [RY00, Lemma 8.5.7]. For r ≥ 2 we have (r −1)p r +1 ≤ ed(PGLpr;p) ≤ p2r−2 +1. ThelowerboundisduetoMerkurjev[Me10b]; theupperboundisprovedina recentpreprintofA.Ruozzi=-=[Ru10]-=-. (Aweakerupperbound,ed(PGLn;p) ≤ 2p2r−2−pr +1, is proved in [MR09b].) In particular, ed(PGLp2;p) = p2 +1; see [Me10a]. Note that the argument in [Me10b] shows that if A is a generic (Z/pZ) rcrossed p... |