Asher Auel

Open problems on central simple algebras

We provide a survey of past research and a list of open problems regarding central simple algebras and the Brauer group over a field, intended both for experts and for beginners.

Fibrations in complete intersections of quadrics, Clifford algebras, derived categories, and rationality problems

Abstract Let X → Y be a fibration whose fibers are complete intersections of r quadrics. We develop new categorical and algebraic tools—a theory of relative homological projective duality and the Morita invariance of the even Clifford algebra under quadric reduction by hyperbolic splitting—to study semiorthogonal decompositions of the bounded derived category D b ( X ) . Together with results in the theory of quadratic forms, we apply these tools in the case where r = 2 and X → Y has relative dimension 1, 2, or 3, in which case the fibers are curves of genus one, Del Pezzo surfaces of degree 4, or Fano threefolds, respectively. In the latter two cases, if Y = P 1 over an algebraically closed field of characteristic zero, we relate rationality questions to categorical representability of X.

Universal Unramified Cohomology of Cubic Fourfolds Containing a Plane

We prove the universal triviality of the third unramified cohomology group of a very general complex cubic fourfold containing a plane. The proof uses results on the unramified cohomology of quadrics due to Kahn, Rost, and Sujatha.

CUBIC FOURFOLDS CONTAINING A PLANE AND A QUINTIC DEL PEZZO SURFACE

We isolate a class of smooth rational cubic fourfolds X containing a plane whose associated quadric surface bundle does not have a rational section. This is equivalent to the nontriviality of the Brauer class of the even Cliord algebra over the K3 surface S of degree 2 arising from X. Specically, we show that in the moduli space of cubic fourfolds, the intersection of divisorsC8\C14 has ve irreducible components. In the component corresponding to the existence of a tangent conic, we prove that the general member is both pfaan and has nontrivial. Such cubic fourfolds provide twisted derived equivalences between K3 surfaces of degrees 2 and 14, hence further corroboration of Kuznetsov’s derived categorical conjecture on the rationality of cubic fourfolds.

Semiorthogonal decompositions and birational geometry of del Pezzo surfaces over arbitrary fields

We study the birational properties of geometrically rational surfaces from a derived categorical perspective. In particular, we give a criterion for the rationality of a del Pezzo surface S over an arbitrary field, namely, that its derived category decomposes into zero-dimensional components. When S has degree at least 5 we construct explicit semiorthogonal decompositions by subcategories of modules over semisimple algebras arising as endomorphism algebras of vector bundles and we show how to retrieve information about the index of S from Brauer classes and Chern classes associated to these vector bundles.

Surjectivity of the total Clifford invariant and Brauer dimension

Abstract A celebrated theorem of Merkurjev—that the 2-torsion of the Brauer group is represented by Clifford algebras of quadratic forms—is in general false when the base is no longer a field. The first counterexamples, when the base is among certain arithmetically subtle hyperelliptic curves over local fields, were constructed by Parimala, Scharlau, and Sridharan. We prove that considering Clifford algebras of all line bundle-valued quadratic forms, such counterexamples disappear and we recover Merkurjevs theorem in these cases: for any smooth curve over a local field or any smooth surface over a finite field, the 2-torsion of the Brauer group is always represented by Clifford algebras of line bundle-valued quadratic forms.

The transcendental lattice of the sextic Fermat surface

We prove that the integral polarized Hodge structure on the transcendental lattice of a sextic Fermat surface is decomposable. This disproves a conjecture of Kulikov related to a Hodge theoretic approach to proving the irrationality of the very general cubic fourfold.

Exceptional collections of line bundles on projective homogeneous varieties

Abstract We construct new examples of exceptional collections of line bundles on the variety of Borel subgroups of a split semisimple linear algebraic group G of rank 2 over a field. We exhibit exceptional collections of the expected length for types A 2 and B 2 = C 2 and prove that no such collection exists for type G 2 . This settles the question of the existence of full exceptional collections of line bundles on projective homogeneous G -varieties for split linear algebraic groups G of rank at most 2.