# Birational geometry of quadrics

Bulletin de la Société Mathématique de France (2009)

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

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topTotaro, 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 -

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