Birational geometry of quadrics

Burt Totaro

Bulletin de la Société Mathématique de France (2009)

  • Volume: 137, Issue: 2, page 253-276
  • ISSN: 0037-9484

Abstract

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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. 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.

How to cite

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Totaro, Burt. "Birational geometry of quadrics." Bulletin de la Société Mathématique de France 137.2 (2009): 253-276. <http://eudml.org/doc/272453>.

@article{Totaro2009,
abstract = {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.},
author = {Totaro, Burt},
journal = {Bulletin de la Société Mathématique de France},
keywords = {quadratic forms; ruled varieties; birational geometry; quadratic Zariski problem},
language = {eng},
number = {2},
pages = {253-276},
publisher = {Société mathématique de France},
title = {Birational geometry of quadrics},
url = {http://eudml.org/doc/272453},
volume = {137},
year = {2009},
}

TY - JOUR
AU - Totaro, Burt
TI - Birational geometry of quadrics
JO - Bulletin de la Société Mathématique de France
PY - 2009
PB - Société mathématique de France
VL - 137
IS - 2
SP - 253
EP - 276
AB - 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.
LA - eng
KW - quadratic forms; ruled varieties; birational geometry; quadratic Zariski problem
UR - http://eudml.org/doc/272453
ER -

References

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