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Birational geometry of quadrics

Burt Totaro — 2009

Bulletin de la Société Mathématique de France

We construct new birational maps between quadrics over a field. The maps apply to several types of quadratic forms, including Pfister neighbors, neighbors of multiples of a Pfister form, and half-neighbors. One application is to determine which quadrics over a field are ruled (that is, birational to the projective line times some variety) in a larger range of dimensions. We describe ruledness completely for quadratic forms of odd dimension at most 17, even dimension at most 10, or dimension 14....

Pseudo-abelian varieties

Burt Totaro — 2013

Annales scientifiques de l'École Normale Supérieure

Chevalley’s theorem states that every smooth connected algebraic group over a perfect field is an extension of an abelian variety by a smooth connected affine group. That fails when the base field is not perfect. We define a pseudo-abelian variety over an arbitrary field k to be a smooth connected k -group in which every smooth connected affine normal k -subgroup is trivial. This gives a new point of view on the classification of algebraic groups: every smooth connected group over a field is an extension...

Line bundles with partially vanishing cohomology

Burt Totaro — 2013

Journal of the European Mathematical Society

Define a line bundle L on a projective variety to be q -ample, for a natural number q , if tensoring with high powers of L kills coherent sheaf cohomology above dimension q . Thus 0-ampleness is the usual notion of ampleness. We show that q -ampleness of a line bundle on a projective variety in characteristic zero is equivalent to the vanishing of an explicit finite list of cohomology groups. It follows that q -ampleness is a Zariski open condition, which is not clear from the definition.

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