arXiv subjects

Daniel Krashen

Publications and source records attributed to Daniel Krashen.

41 records · Page 3Linked to original sources

arXiv2006

Zero cycles on homogeneous varieties

In this paper we study the group $A_0(X)$ of zero dimensional cycles of degree 0 modulo rational equivalence on a projective homogeneous algebraic variety $X$. To do this we translate rational equivalence of 0-cycles on a projective variety into R-equivalence on symmetric powers of the variety. For certain homogeneous varieties, we then relate these symmetric powers to moduli spaces of étale subalgebras of central simple algebras which we construct. This allows us to show $A_0(X) = 0$ for certain classes of homogeneous varieties, extending previous results of Swan / Karpenko, of Merkurjev, and of Panin.

math.AG↗

arXiv2006

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. In this paper, we exhibit a relationship between motives of two homogeneous varieties by producing a natural rational map between them which becomes a projective bundle morphism after being resolved. This allows one to use formulas for projective bundles and blowing up to relate the motives of the two varieties. We believe that in the future this ideas could be used to discover more relationships between other types of homogeneous varieties.

math.AG↗

arXiv2004

Moduli of étale subalgebras in an Azumaya algebra

Let $A$ be a sheaf of Azumaya algebras over a Noetherian base $S$. In this paper we describe using generalized Severi-Brauer varieties, a quasi-projective moduli space parametrizing sheaves of ètale subalgebras of $A$. In the case that $S$ is the spectrum of a field, we study the geometry of this moduli space, and show that in certain cases it is a rational variety.

math.RA↗

arXiv2002

Severi-Brauer varieties of semidirect product algebras

A conjecture of Amitsur states that two Severi-Brauer varieties are birationally isomorphic if and only if the underlying algebras are the same degree and generate the same cyclic subgroup of the Brauer group. It is known that generating the same cyclic subgroup is a necessary condition, however it has not yet been shown to be sufficient. In this paper we examine the case where the algebras have a maximal subfield $K/F$ of degree $n$ with Galois closure $E/F$ whose Galois group is of the form $C_n \rtimes H$, where $E^H = K$ and $|H|$ is prime to $n$. For such as we show that the conjecture is true for certain cases of $n$ and $H$. In particular we prove the conjecture in the case that $G$ is a dihedral group of order $2p$, where $p$ is prime

math.RA↗

arXiv2002

Birational isomorphisms between generalized Severi-Brauer varieties

The aim of this paper is to investigate the birational geometry of Generalized Severi-Brauer varieties. A conjecture of Amitsur states that two Severi-Brauer varieties $V(A)$ and $V(B)$ are birational if the underlying central simple algebras $A$ and $B$ are the same degree and generate the same cyclic subgroup of the Brauer group. We present a generalization of this conjecture to Generalized Severi-Brauer varieties, and show that in most cases we may reduce the new conjecture to the case where every subfield of the algebras is maximal, and in particular to the case where the algebras have prime power degree. This allows us to prove infinitely many new cases for Amitsur's original conjecture. We also give a proof of the generalized conjecture for the case $B \cong A^{op}$.

math.RA↗