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Birational Finite Extensions of Mappings from a Smooth Variety

Marek Karaś (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

We present an example of finite mappings of algebraic varieties f:V → W, where V ⊂ kⁿ, W ⊂ k n + 1 , and F : k ⁿ → k n + 1 such that F | V = f and gdeg F = 1 < gdeg f (gdeg h means the number of points in the generic fiber of h). Thus, in some sense, the result of this note improves our result in J. Pure Appl. Algebra 148 (2000) where it was shown that this phenomenon can occur when V ⊂ kⁿ, W ⊂ k m with m ≥ n+2. In the case V,W ⊂ kⁿ a similar example does not exist.

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....

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