Severi-Brauer Varieties Research Articles

Absolutely split locally free sheaves on proper k-schemes and Brauer–Severi varieties

We classify absolutely split vector bundles on proper k-schemes. More precise, we prove that the closed points of the Picard scheme are in one-to-one correspondence with indecomposable absolutely split vector bundles. Furthermore, we apply the obtained results to study the geometry of (generalized) Brauer–Severi varieties.

Ulrich bundles on Brauer–Severi varieties

We prove the existence of Ulrich bundles on any Brauer–Severi variety. In some cases, the minimal possible rank of the obtained Ulrich bundles equals the period of the Brauer–Severi variety. Moreover, we find a formula for the rank of an Ulrich bundle involving the period of the considered Brauer–Severi variety X X , at least if d i m ( X ) = p − 1 \mathrm {dim}(X)=p-1 for an odd prime p p . This formula implies that the rank of any Ulrich bundle on such a Brauer–Severi variety X X must be a multiple of the period.

On the Chow groups of a biquaternion Severi–Brauer variety

We provide an alternative proof that the Chow group of $1$-cycles on a Severi--Brauer variety associated to a biquaternion division algebra is torsion-free. There are three proofs of this result in the literature, all of which are due to Karpenko and rely on a clever use of $K$-theory. The proof that we give here, by contrast, is geometric and uses degenerations of quartic elliptic normal curves.

Jacques Tits motivic measure

In this article we construct a new motivic measure called the Jacques Tits motivic measure. As a first main application, we prove that two Severi-Brauer varieties (or, more generally, two twisted Grassmannian varieties), associated to 2-torsion central simple algebras, have the same class in the Grothendieck ring of varieties if and only if they are isomorphic. In addition, we prove that if two Severi-Brauer varieties, associated to central simple algebras of period {3, 4, 5, 6}, have the same class in the Grothendieck ring of varieties, then they are necessarily birational to each other. As a second main application, we prove that two quadric hypersurfaces (or, more generally, two involution varieties), associated to quadratic forms of dimension 6 or to quadratic forms of arbitrary dimension defined over a base field k with I^3(k)=0, have the same class in the Grothendieck ring of varieties if and only if they are isomorphic. In addition, we prove that the latter main application also holds for products of quadric hypersurfaces.

Addition of Brauer classes via Picard schemes

We give an explicit way to construct a Brauer-Severi variety corresponding to the sum of the two Brauer classes that split by a common Galois extension. We apply the result to explicitly calculate the components of Picard schemes for stable curves with geometrically rational components.

Construction of Brauer-Severi Varieties

In this paper, we give an algorithm for computing equations of Brauer-Severi varieties over fields of characteristic 0. As an example, we show the equations of all Brauer-Severi surfaces defined over Q.

Boundedness for finite subgroups of linear algebraic groups

We show the boundedness of finite subgroups in any anisotropic reductive group over a perfect field that contains all roots of 1 1 . Also, we provide explicit bounds for orders of finite subgroups of automorphism groups of Severi–Brauer varieties and quadrics over such fields.

On the section conjecture and Brauer\u2013Severi varieties

J. Stix proved that a curve of positive genus over mathbb {Q} which maps to a non-trivial Brauer–Severi variety satisfies the section conjecture. We prove that, if X is a curve of positive genus over a number field k and the Weil restriction R_{k/mathbb {Q}}X admits a rational map to a non-trivial Brauer–Severi variety, then X satisfies the section conjecture. As a consequence, if X maps to a Brauer–Severi variety P such that the corestriction {text {cor}}_{k/mathbb {Q}}([P])in {text {Br}}(mathbb {Q}) is non-trivial, then X satisfies the section conjecture.

On cubic torsors, biextensions and Severi–Brauer varieties over Abelian varieties

We study homogeneous irreducible Severi–Brauer varieties over an Abelian variety A. Such objects were classified by Brion (Algebra Number Theory 7(10):2475–2510, 2013). Here we interpret that result within the context of cubic structures and biextensions for certain \(\mathbb {G}_m\)-torsors over finite subgroups of A. Our results build on the theory of Breen (Fonctions thêta et théorème du cube, Springer, Berlin, 1983), and Moret-Bailly (Pinceaux de variétés abéliennes. Astérisque 129:266, 1985).

No phantoms in the derived category of curves over arbitrary fields, and derived characterizations of Brauer–Severi varieties

We show that the derived category of Brauer–Severi curves satisfies the Jordan–Holder property and cannot have quasi-phantoms, phantoms or universal phantoms. In this way we obtain that quasi-phantoms, phantoms or universal phantoms cannot exist in the derived category of smooth projective curves over a field k. Moreover, we show that a n-dimensional Brauer–Severi variety is completely characterized by the existence of a full weak exceptional collection consisting of pure vector bundles of length n+1, at least in characteristic zero. We conjecture that Brauer–Severi varieties X satisfy rdimcat(X)= ind(X)−1, provided period equals index, and prove this in the case of curves, surfaces and for Brauer–Severi varieties of index at most three. We believe that the results for curves are known to the experts. We nevertheless give the proofs, adding to the literature.

Algebraic connective $K$-theory of a Severi-Brauer variety with prescribed reduced behavior

We show that Chow groups of low dimension cycles are torsion free for a class of sufficiently generic Severi-Brauer varieties. Using a recent result of Karpenko, this allows us to compute the algebraic connective K -theory in low degrees for the same class of varieties. Independently of these results, we show that the associated graded ring for the topological filtration on the Grothendieck ring is torsion free in the same degrees for an arbitrary Severi-Brauer variety.

Decompositions of derived categories of gerbes and of families of Brauer-Severi varieties

It is well known that the category of quasi-coherent sheaves on a gerbe banded by a diagonalizable group decomposes according to the characters of the group. We establish the corresponding decomposition of the unbounded derived category of complexes of sheaves with quasi-coherent cohomology. This generalizes earlier work by Lieblich for gerbes over schemes whereas our gerbes may live over arbitrary algebraic stacks. By combining this decomposition with the semi-orthogonal decomposition for a projectivized vector bundle, we deduce a semi-orthogonal decomposition of the derived category of a family of Brauer-Severi varieties whose components can be described in terms of twisted sheaves on the base. This reproves and generalizes a result of Bernardara.

Codimension 2 cycles on Severi–Brauer varieties and decomposability

In this text we show that one can generalize results showing that $$\mathrm {CH}^2(X)$$ , for various Severi–Brauer varieties X, is sometimes torsion free. In particular we show that for any pair of odd integers (n, m), with m dividing n and sharing the same prime factors, one can find a central simple k-algebra A of index n and exponent m that moreover has $$\mathrm {CH}^2(X)$$ torsion free for $$X=\mathrm {SB}(A)$$ . One can even take $$k={\mathbb {Q}}$$ in this construction.

Glider Brauer-Severi varieties of central simple algebras

The glider Brauer-Severi variety GBS(A) of a central simple algebra A over a field K is introduced as the set of all irreducible left glider ideals in A for some filtration FA. For fields we deduce that GBS(K) equals R(K)×Z, the product of the Riemann surface of K and Z. For a csa A over K it turns out that GBS(A)=BS(A)×GBS(K), where BS(A) denotes the classical Brauer-Severi variety of A.

A birational interpretation of Severi-Brauer varieties

Let F be a field, A an F-csa of degree n. Saltman proves that there is a Zariski dense subset of elements such that the Severi-Brauer variety of A is birationally equivalent to the affine variety defined by restricting the reduced norm to subsets of the form K + d where K is a commutative separable maximal F subalgebra of A. In this work, we show that Saltman’s result actually implies that there is a non-empty dense subset of subspaces of dimension n + 1 such that the Severi-Brauer variety of A is birationally equivalent to the projective variety defined by restricting the reduced norm of A to V. We then show that for symbol algebras, a standard n-Kummer subspace induces the birational equivalence as above and use it to reprove Amitsur’s conjecture for the case of symbol algebras.

On the K-theory coniveau epimorphism for products of Severi–Brauer varieties

For X a product of Severi–Brauer varieties, we conjecture that if the Chow ring of X is generated by Chern classes, then the canonical epimorphism from the Chow ring of X to the graded ring associated to the coniveau filtration of the Grothendieck ring of X is an isomorphism. We show this conjecture is equivalent to the condition that if G is a split semisimple algebraic group of type AC, B is a Borel subgroup of G and E is a standard generic G-torsor, then the canonical epimorphism from the Chow ring of E∕B to the graded ring associated with the coniveau filtration of the Grothendieck ring of E∕B is an isomorphism. In certain cases we verify this conjecture.

Derived Azumaya algebras and twisted K-theory

We construct a relative version of topological K-theory, in the sense of [11], over an arbitrary quasi-compact, quasi-separated C-scheme X. This has as input a Perf(X)-linear stable ∞-category and output a sheaf of spectra on X(C), the space of complex points of X. We then characterize the values of this functor on inputs of the form Perf(X,A), for A a derived Azumaya algebra over X. In such cases we show that this coincides with the α-twisted topological K-theory of X(C) for some appropriately defined twist of K-theory. We use this to provide a topological analogue of a classical result of Quillen's on the algebraic K-theory of Severi-Brauer varieties.

The Picard group of Brauer-Severi varieties

AbstractIn this paper, we provide explicit generators for the Picard groups of cyclic Brauer-Severi varieties defined over the base field. In particular,we provide such generators for all Brauer-Severi surfaces. To produce these generators we use the theory of twists of smooth plane curves.

The Coniveau Filtration on for Some Severi–Brauer Varieties

AbstractWe produce an isomorphism $E_{\infty }^{m,-m-1}\cong \text{Nrd}_{1}(A^{\otimes m})$ between terms of the $\text{K}$-theory coniveau spectral sequence of a Severi–Brauer variety $X$ associated with a central simple algebra $A$ and a reduced norm group, assuming $A$ has equal index and exponent over all finite extensions of its center and that $\text{SK}_{1}(A^{\otimes i})=1$ for all $i>0$.

Orthogonal involutions on central simple algebras and function fields of Severi–Brauer varieties

An orthogonal involution σ on a central simple algebra A, after scalar extension to the function field F(A) of the Severi–Brauer variety of A, is adjoint to a quadratic form qσ over F(A), which is uniquely defined up to a scalar factor. Some properties of the involution, such as hyperbolicity, and isotropy up to an odd-degree extension of the base field, are encoded in this quadratic form, meaning that they hold for the involution σ if and only if they hold for qσ. As opposed to this, we prove that there exists non-totally decomposable orthogonal involutions that become totally decomposable over F(A), so that the associated form qσ is a Pfister form. We also provide examples of nonisomorphic involutions on an index 2 algebra that yield similar quadratic forms, thus proving that the form qσ does not determine the isomorphism class of σ, even when the underlying algebra has index 2. As a consequence, we show that the e3 invariant for orthogonal involutions is not classifying in degree 12, and does not detect totally decomposable involutions in degree 16, as opposed to what happens for quadratic forms.