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Rational maps between quasilinear hypersurfaces

Part of: Birational geometry Forms and linear algebraic groups Basic linear algebra

Published online by Cambridge University Press:  17 December 2012

Stephen Scully
Stephen Scully*
Affiliation:
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK (email: [email protected])
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Abstract

We prove analogues of several well-known results concerning rational maps between quadrics for the class of so-called quasilinear p-hypersurfaces. These hypersurfaces are nowhere smooth over the base field, so many of the geometric methods which have been successfully applied to the study of projective homogeneous varieties over fields cannot be used. We are therefore forced to take an alternative approach, which is partly facilitated by the appearance of several non-traditional features in the study of these objects from an algebraic perspective. Our main results were previously known for the class of quasilinear quadrics. We provide new proofs here, because the original proofs do not immediately generalise for quasilinear hypersurfaces of higher degree.

Keywords

quasilinear formsquasilinear hypersurfacesrational mapsbirational geometry

MSC classification

Secondary: 11E04: Quadratic forms over general fields 11E76: Forms of degree higher than two 14E05: Rational and birational maps 15A03: Vector spaces, linear dependence, rank
Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 3 , March 2013 , pp. 333 - 355
Copyright
Copyright © 2012 The Author(s)

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References

[BK86]Bloch, S. and Kato, K., p-adic étale cohomology, Publ. Math. Inst. Hautes Études Sci. (1986), 107–152.CrossRefGoogle Scholar
[EKM08]Elman, R., Karpenko, N. and Merkurjev, A., The algebraic and geometric theory of quadratic forms, American Mathematical Society Colloquium Publications, vol. 56 (American Mathematical Society, Providence, RI, 2008).CrossRefGoogle Scholar
[Hau12]Haution, O., Integrality of the Chern character in small codimension, Adv. Math. 231 (2012), 855–878.CrossRefGoogle Scholar
[Hof95]Hoffmann, D. W., Isotropy of quadratic forms over the function field of a quadric, Math. Z. 220 (1995), 461–476.CrossRefGoogle Scholar
[Hof04]Hoffmann, D. W., Diagonal forms of degree p in characteristic p, in Algebraic and arithmetic theory of quadratic forms, Contemporary Mathematics, vol. 344 (American Mathematical Society, Providence, RI, 2004), 135–183.CrossRefGoogle Scholar
[HI00]Hoffmann, D. W. and Izhboldin, O. T., Embeddability of quadratic forms in Pfister forms, Indag. Math. (N.S.) 11 (2000), 219–237.CrossRefGoogle Scholar
[HL04]Hoffmann, D. W. and Laghribi, A., Quadratic forms and Pfister neighbors in characteristic 2, Trans. Amer. Math. Soc. 356 (2004), 4019–4053.CrossRefGoogle Scholar
[HL06]Hoffmann, D. W. and Laghribi, A., Isotropy of quadratic forms over the function field of a quadric in characteristic 2, J. Algebra 295 (2006), 362–386.CrossRefGoogle Scholar
[Izh00]Izhboldin, O. T., Motivic equivalence of quadratic forms II, Manuscripta Math. 102 (2000), 41–52.CrossRefGoogle Scholar
[Kar00]Karpenko, N. A., On anisotropy of orthogonal involutions, J. Ramanujan Math. Soc. 15 (2000), 1–22.Google Scholar
[Kar03]Karpenko, N. A., On the first Witt index of quadratic forms, Invent. Math. 153 (2003), 455–462.CrossRefGoogle Scholar
[Kat82]Kato, K., Galois cohomology of complete discrete valuation fields, in Algebraic K-theory, Part II (Oberwolfach, 1980), Lecture Notes in Mathematics, vol. 967 (Springer, Berlin, 1982), 215–238.CrossRefGoogle Scholar
[KM03]Karpenko, N. and Merkurjev, A., Essential dimension of quadrics, Invent. Math. 153 (2003), 361–372.CrossRefGoogle Scholar
[Lan02]Lang, S., Algebra, Graduate Texts in Mathematics, vol. 211, revised third edition (Springer, Berlin, 2002).CrossRefGoogle Scholar
[LM07]Levine, M. and Morel, F., Algebraic cobordism, in Springer monographs in mathematics (Springer, Berlin, 2007).Google Scholar
[Mer00]Merkurjev, A., Degree formula, Preprint (2000), available at http://www.math.uni-bielefeld.de/∼rost/degree-formula.html.Google Scholar
[Mer03]Merkurjev, A., Steenrod operations and degree formulas, J. Reine Angew. Math. 565 (2003), 13–26.Google Scholar
[Sch10]Schröer, S., On fibrations whose geometric fibers are nonreduced, Nagoya Math. J. 200 (2010), 35–57.CrossRefGoogle Scholar
[Tot07]Totaro, B., The automorphism group of an affine quadric, Math. Proc. Cambridge Philos. Soc. 143 (2007), 1–8.CrossRefGoogle Scholar
[Tot08]Totaro, B., Birational geometry of quadrics in characteristic 2, J. Algebraic Geom. 17 (2008), 577–597.CrossRefGoogle Scholar
[Tot09]Totaro, B., Birational geometry of quadrics, Bull. Soc. Math. France 137 (2009), 253–276.CrossRefGoogle Scholar
[Vis99]Vishik, A., Direct summands in the motives of quadrics, Preprint (1999), available at http://www.maths.nott.ac.uk/personal/av/papers.html.Google Scholar
[Zai10]Zainoulline, K., Degree formula for connective K-theory, Invent. Math. 179 (2010), 507–522.CrossRefGoogle Scholar