Severi-Brauer Varieties Research Articles

Absolutely split locally free sheaves on Brauer–Severi varieties of index two

Let X be a Brauer–Severi variety over a field k associated with a central simple k -algebra of index two. This variety has the property of being isomorphic to a projective space P L N after base change to a degree two Galois extension L . A locally free sheaf on X is called absolutely split if it splits after base change as a direct sum of invertible sheaves on the projective space. We classify the isomorphism classes of absolutely split locally free sheaves on X .

Genus one curves and Brauer–Severi varieties

Genus one curves and Brauer–Severi varieties

Decompositions of motives of generalized Severi-Brauer varieties

Let p be a positive prime number and X be a Severi-Brauer variety of a central division algebra D of degree p^n , with n\geq 1 . We describe all shifts of the motive of X in the complete motivic decomposition of a variety Y , which splits over the function field of X and satisfies the nilpotence principle. In particular, we prove the motivic decomposability of generalized Severi-Brauer varieties X(p^m,D) of right ideals in D of reduced dimension p^m, m=0,1,ldots,n-1 , except the cases p=2, m=1 and m=0 (for any prime p) , where motivic indecomposability was proven by Nikita Karpenko.

Incompressibility of quadratic Weil transfer of generalized Severi–Brauer varieties

AbstractLet X be the variety obtained by the Weil transfer with respect to a quadratic separable field extension of a generalized Severi–Brauer variety. We study (and, in some cases, determine) the canonical dimension, incompressibility, and motivic indecomposability of X. We determine the canonical 2-dimension of X (in the general case).

Cohomologie non ramifiée en degré trois dʼune variété de Severi–Brauer

Let S be a smooth projective geometrically integral surface defined over a finite field F , char. F ≠ 2 , and let K be its field of fractions. Parimala and Suresh (2010) [9] proved that for C a conic over K, the group H nr 3 ( K ( C ) / F , Q ℓ / Z ℓ ( 2 ) ) is zero for ℓ ≠ char. F . In this Note we extend their result to the case of Severi–Brauer varieties of prime index.

Brauer group of a moduli space of parabolic vector bundles over a curve

AbstractLet be a moduli space of stable parabolic vector bundles of rank n ≥ 2 and fixed determinant of degree d over a compact connected Riemann surface X of genus g(X) ≥ g(X) = 2, then we assume that n > 2. Let m denote the greatest common divisor of d, n and the dimensions of all the successive quotients of the quasi–parabolic filtrations. We prove that the Brauer group Br is isomorphic to the cyclic group ℤ/mℤ. We also show that Br is generated by the Brauer class of the Brauer–Severi variety over obtained by restricting the universal projective bundle over X × .

Thin Severi-Brauer Varieties

Severi-Brauer varieties are twisted forms of projective spaces (in the sense of Galois cohomology) and are associated in a functorial way to central simple algebras. Similarly quadrics are related to algebras with involution. Since thin projective spaces are finite sets, thin Severi-Brauer varieties are finite sets endowed with a Galois action; they are associated to etale algebras. Similarly, thin quadrics are etale algebras with involution. We discuss embeddings of thin Severi-Brauer varieties and thin quadrics in Severi-Brauer varieties and quadrics as geometric analogues of embeddings of etale algebras into central simple algebras (with or without involution), and consider the geometric counterpart of the Clifford algebra construction.

Degeneracy and decomposability in abelian crossed products

In this paper we continue the study of the relationship between degeneracy and decomposability in abelian crossed products (K. McKinnie (2008) [McK08]). In particular we construct an indecomposable abelian crossed product division algebra of exponent p and index p 2 for p an odd prime. The algebra we construct is generic in the sense of S.A. Amitsur and D. Saltman (1978) [AS78] and has the property that its underlying abelian crossed product is a decomposable division algebra defined by a non-degenerate matrix. This algebra gives an example of an indecomposable generic abelian crossed product which is shown to be indecomposable without using torsion in the Chow group of the corresponding Severi–Brauer variety as was needed in N.A. Karpenko (1998) [Kar98] and K. McKinnie (2008) [McK08]. It also gives an example of a Brauer class which is in Tignol's Dec group with respect to one abelian maximal subfield, but not in the Dec group with respect to another.

Motivic decomposability of generalized Severi–Brauer varieties

Let F be an arbitrary field. Let p be a positive prime number and D a central division F-algebra of degree p n , with n ⩾ 1 . We write SB ( p m , D ) for the generalized Severi–Brauer variety of right ideals in D of reduced dimension p m for m = 0 , 1 , … , n − 1 . We note by M ( SB ( p m , D ) ) the Chow motive with coefficients in F p of the variety SB ( p m , D ) . It was proven by Nikita Karpenko that this motive is indecomposable for any prime p and m = 0 and for p = 2 , m = 1 . We prove decomposability of M ( SB ( p m , D ) ) in all the other cases.

Witt groups of hermitian forms over a Brauer–Severi variety

Let X be a Brauer-Severi variety over a field k of characteristic not 2, and let D be a division algebra over k with a k-linear involution. We investigate Witt groups of certain hermitian forms over D ⊗ k O X .

Motives of Azumaya algebras

AbstractWe study the slice filtration for theK-theory of a sheaf of Azumaya algebrasA, and for the motive of a Severi-Brauer variety, the latter in the case of a central simple algebra of prime degree over a field. Using the Beilinson–Lichtenbaum conjecture, we apply our results to show the vanishing ofSK2(A) for a central simple algebraAof square-free index (prime to the characteristic). This proves a conjecture of Merkurjev.

Jordan Algebras Over Algebraic Varieties

General results on the module structure of Jordan algebras over locally ringed spaces are obtained. Albert algebras over a Brauer–Severi variety with associated central simple algebra of degree 3 are constructed using generalizations of the Tits process and the first Tits construction.

Indecomposable p-algebras and Galois subfields in generic abelian crossed products

Let F be a Henselian valued field with char ( F ) = p and D a semi-ramified, “not strongly degenerate” p-algebra. We show that all Galois subfields of D are inertial. Using this as a tool we study generic abelian crossed product p-algebras, proving among other things that the noncyclic generic abelian crossed product p-algebras defined by non-degenerate matrices are indecomposable p-algebras. To construct examples of these indecomposable p-algebras with exponent p and large index we study the relationship between degeneracy in matrices defining abelian crossed products and torsion in CH 2 of Severi–Brauer varieties.

Birational maps between generalized Severi–Brauer varieties

A conjecture of Amitsur states that two Severi–Brauer varieties V ( A ) and V ( B ) are birationally isomorphic if and only if the underlying algebras A and B are the same degree and generate the same cyclic subgroup of the Brauer group. We examine the question of finding birational isomorphisms between generalized Severi–Brauer varieties. As a first step, we exhibit a birational isomorphism between the generalized Severi–Brauer variety of an algebra and its opposite. We also extend a theorem of P. Roquette to generalized Severi–Brauer varieties and use this to show that one may often reduce the problem of finding birational isomorphisms to the case where each of the separable subfields of the corresponding algebras are maximal, and therefore to the case where the algebras have prime power degree. We observe that this fact allows us to verify Amitsur’s conjecture for many particular cases.

Motives of unitary and orthogonal homogeneous varieties

Grothendieck–Chow motives of quadric hypersurfaces have provided many insights into the theory of quadratic forms. Subsequently, the landscape of motives of more general projective homogeneous varieties has begun to emerge. In particular, there have been many results which relate the motive of a one homogeneous variety to motives of other simpler or smaller ones (see for example [N.A. Karpenko, Cohomology of relative cellular spaces and of isotropic flag varieties, Algebra i Analiz 12 (1) (2000) 3–69. [Kar00a]; V. Chernousov, S. Gille, A. Merkurjev, Motivic decomposition of isotropic projective homogeneous varieties, Duke Math. J. 126 (1) (2005) 137–159. [CGM05]; P. Brosnan, On motivic decompositions arising from the method of Białynicki-Birula, Invent. Math. 161 (1) (2005) 91–111. [Bro05]; S. Nikolenko, N. Semenov, K. Zainoulline, Motivic decomposition of anisotropic varieties of type F 4 into generalized Rost motives, preprint, Max-Planck-Institut für Mathematik, 90, 2005. [NSZ05]; K.V. Zaĭnullin, N.S. Semenov, On the classification of projective homogeneous varieties up to motivic isomorphism, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 330 (2006) 158–172, 273. [ZS06]; B. Calmès, V. Petrov, N. Semenov, K. Zainoulline, Chow motives of twisted flag varieties, Compos. Math. 142 (4) (2006) 1063–1080. [CPSZ06]; K. Zainoulline, Motivic decomposition of a generalized Severi–Brauer variety, arXiv: math.AG/0601666. [Zai]]). In this paper, we exhibit a relationship between motives of two homogeneous varieties by producing a natural rational map between them. As an application, we compute the Chow group of zero-dimensional cycles on a homogeneous variety associated to a Hermitian form.

Noncommutative cyclic covers and maximal orders on surfaces

In this paper, we construct various examples of maximal orders on surfaces, including some del Pezzo orders, some ruled orders and some numerically Calabi–Yau orders. The method of construction is a noncommutative version of the cyclic covering trick. These noncommutative cyclic covers are very computable and we give a formula for their ramification data. This often allows us to determine if a maximal order, described via ramification data, can be constructed as a noncommutative cyclic cover. The construction also has applications to Brauer–Severi varieties and, in the quaternion case, we show how to obtain some Brauer–Severi varieties from G-Hilbert schemes of P 1 -bundles.

Bicyclic algebras of prime exponent over function fields

We examine some properties of bicyclic algebras, i.e. the tensor product of two cyclic algebras, defined over a purely transcendental function field in one variable. We focus on the following problem: When does the set of local invariants of such an algebra coincide with the set of local invariants of some cyclic algebra? Although we show this is not always the case, we determine when it happens for the case where all degeneration points are defined over the ground field. Our main tool is Faddeev's theory. We also study a geometric counterpart of this problem (pencils of Severi-Brauer varieties with prescribed degeneration data).

The normalization constant of a certain invariant measure on GLn()

The ratio of the Tamagawa measure and a certain invariant measure on the group GL n ( Open image in new window ) is computed, where Open image in new window is the adele of a division algebra D over a global field. An explicit formula of the ratio is described in terms of the special values of the zeta function of D. This formula yields (i) an explicit lower bound of the Hermite–Rankin constant γ n,m (D) of D and (ii) an explicit asymptotic behavior of the distribution of rational points on Brauer–Severi variety.

Quaternionic algebraic cycles and reality

In this paper we compute the equivariant homotopy type of spaces of algebraic cycles on real Brauer-Severi varieties, under the action of the Galois group G a l ( C / R ) Gal({\mathbb C} / {\mathbb R}) . Appropriate stabilizations of these spaces yield two equivariant spectra. The first one classifies Dupont/Seymour’s quaternionic K K -theory, and the other one classifies an equivariant cohomology theory Z ∗ ( − ) {\mathfrak Z}^*(-) which is a natural recipient of characteristic classes K H ∗ ( X ) → Z ∗ ( X ) KH^*(X) \to {\mathfrak Z}^*(X) for quaternionic bundles over Real spaces X X .

The Brauer group and the second Abel-Jacobi map for 0-cycles on algebraic varieties

This paper focuses on the connection between the Brauer group and the $0$-cycles of an algebraic variety. We give an alternative construction of the second $l$-adic Abel-Jacobi map for such cycles, linked to the algebraic geometry of Severi-Brauer varieties on $X$. This allows us then to relate this Abel-Jacobi map to the standard pairing between $0$-cycles and Brauer groups (see [M], [L]), completing results from [M] in this direction. Second, for surfaces, it allows us to present this map according to the more geometrical approach devised by M. Green in the framework of (arithmetic) mixed Hodge structures (see [G]). Needless to say, this paper owes much to the work of U. Jannsen and, especially, to his recently published older letter [J4] to B. Gross.