Absolutely split locally free sheaves on proper kk-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.Comment: 17 pages, revised version with corrected and improved result, comments are welcome

Rational maps between varieties associated to central simple algebras

In this paper we show that if two central simple $k$-algebras generate the same cyclic subgroup in $\mathrm{Br}(k)$, then there are rational maps between varieties associated to these algebras, such as Brauer--Severi varieties, norm hypersurfaces and symmetric powers. In some cases we even have rational embeddings. We also relate the obtained results to the Amitsur conjecture.Comment: 15 pages, comments are welcom

Non-existence of exceptional collections on twisted flags and categorical representability via noncommutative motives

In this paper we prove that a finite product of Brauer--Severi varieties is categorical representable in dimension zero if and only if it admits a $k$-rational point if and only if it is rational over $k$. The same is true for certain isotropic involution varieties over a field $k$ of characteristic different from two. For finite products of generalized Brauer--Severi varieties, categorical representability in dimension zero is equivalent to the existence of a full exceptional collection. In this case however categorical representability in dimension zero is not equivalent to the existence of a rational point. We also show that non-trivial twisted flags of classical type $A_n$ and $C_n$ cannot have full exceptional collections, enlarging in this way the set of previous known examples. Finally, we determine the categorical representability dimension $\mathrm{rdim}(X)$ for generalized Brauer--Severi varieties of index $\leq 3$ and for certain twisted forms of smooth quadrics (involution varieties).Comment: 21 pages, revised version with added results, comments are welcome

Rational points on symmetric powers and categorical representability

In this paper we observe that for geometrically integral projective varieties $X$, admitting a full weak exceptional collection consisting of pure vector bundles, the existence of a $k$-rational point implies $\mathrm{rdim}(X)=0$. We also study the symmetric power $S^n(X)$ of Brauer--Severi and involution varieties over $\mathbb{R}$ and prove that the equivariant derived category $D^b_{S_n}(X^n)$ admits a full weak exceptional collection. As a consequence, we find $\mathrm{rdim}(X)=0$ if and only if $\mathrm{rdim}(D^b_{S_n}(X^n))=0$ for $1\leq n\leq 3$. If $X$ is Brauer--Severi, the existence of a $\mathbb{R}$-rational point on $X$ or $S^3(X)$ is equivalent to $\mathrm{rdim}(D^b_{S_3}(X^3))=0$.Comment: 15 pages, revised version with added results, comments are welcome. arXiv admin note: text overlap with arXiv:1607.0104

On non-existence of full exceptional collections on some relative flags

In this paper we show that certain relative flags cannot have full exceptional collections. We also prove that some of these flags are categorical representable in dimension zero if and only if they admit a full exceptional collection. As a consequence, these flags are categorical representable in dimension zero if and only if they have $k$-rational points if and only if they are $k$-rational. Moreover, we calculate the categorical representability dimension for the flags under consideration.Comment: 10 pages, revised version with added results, comments are welcome! arXiv admin note: text overlap with arXiv:1607.0104

Tilting objects on twisted forms of some relative flag varieties

We prove the existence of tilting objects on generalized Brauer--Severi varieties, some relative flags and some twisted forms of relative flags. As an application we obtain tilting objects on certain homogeneous varieties of classical type and on certain twisted forms of homogeneous varieties of type $A_n$ and $C_n$

No full exceptional collections on non-split Brauer--Severi varieties of dimension ≤3\leq 3

In an earlier paper we showed that non-split Brauer--Severi curves do not admit full strong exceptional collections. In the present note we extend this observation and prove that there cannot exist full exceptional collections on non-split Brauer--Severi varieties of dimension $\leq 3$.Comment: commetnts welcom

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 $X$, at least if $\mathrm{dim}(X)=p-1$ for an odd prime $p$. This formula implies that the rank of any Ulrich bundle on such a Brauer--Severi variety $X$ must be a multiple of the period.Comment: 14 pages, to appear in Proceedings of the American Mathematical Society, comments welcome

Tilting objects on some global quotient stacks

We prove the existence of tilting objects on some global quotient stacks. As a consequence we provide further evidence for a conjecture on the Rouquier dimension of derived categories formulated by Orlov.Comment: revised version, To appear in J. Commut. Algebra. arXiv admin note: text overlap with arXiv:1511.0700

Batched computation of the singular value decompositions of order two by the AVX-512 vectorization

In this paper a vectorized algorithm for simultaneously computing up to eight singular value decompositions (SVDs, each of the form $A=U\Sigma V^{\ast}$) of real or complex matrices of order two is proposed. The algorithm extends to a batch of matrices of an arbitrary length $n$, that arises, for example, in the annihilation part of the parallel Kogbetliantz algorithm for the SVD of a square matrix of order $2n$. The SVD algorithm for a single matrix of order two is derived first. It scales, in most instances error-free, the input matrix $A$ such that its singular values $\Sigma_{ii}$ cannot overflow whenever its elements are finite, and then computes the URV factorization of the scaled matrix, followed by the SVD of a non-negative upper-triangular middle factor. A vector-friendly data layout for the batch is then introduced, where the same-indexed elements of each of the input and the output matrices form vectors, and the algorithm's steps over such vectors are described. The vectorized approach is then shown to be about three times faster than processing each matrix in isolation, while slightly improving accuracy over the straightforward method for the $2\times 2$ SVD.Comment: Preprint of an article submitted for consideration in Parallel Processing Letters ( https://www.worldscientific.com/worldscinet/ppl