### Projective resolutions of representations of GL (n).

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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....

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...

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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