Burt Totaro: Birational geometry of quadrics


Submission: Primary 11E04, Secondary 14E05.

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. The proof uses a new structure theorem for 14-dimensional forms, generalizing Izhboldin's theorem on 10-dimensional forms. We also show that Vishik's 16-dimensional form is ruled.

2000 Mathematics Subject Classification: Primary 11E04, Secondary 14E05.

Keywords and Phrases: Quadratic forms, ruled varieties, birational geometry, quadratic Zariski problem

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