New York Journal of Mathematics
Volume 30 (2024), 550-560


Nikita A. Karpenko

On Spin(2023)-torsors

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Published: April 5, 2024.
Keywords: Quadratic forms over fields; affine algebraic groups; spin groups; projective homogeneous varieties; Chow rings.
Subject [2020]: 20G15; 14C25

The torsion index of a spin group Spin(d), describing the splitting behaviour of generic Spin(d)-torsor E, is a 2-power 2t with the torsion exponent t determined by B. Totaro in 2005. The critical exponent it is responsible for partial splitting behaviour of E and takes values inside the doubleton {t-1, t}. For all d less than or equal to 16, the value of it is known to be high. The very first case of the low value, obtained very recently, is d=17. In the present work, we develop a new method which allows one to show that it=t-1 for most d. In particular, it is shown that it is low for every d=2r+1 with r greater than or equal to 4 as well as for d=2023, playing the role of a "randomly chosen" high dimension. For d=2023, using an extension of the new method (applicable to arbitrary d), several exponents beyond the critical one are also determined.


The author's work has been supported by a Discovery Grant from the National Science and Engineering Research Council of Canada. It has been finalized during his stay at the Universite de Lorraine.

Author information

Nikita A. Karpenko
Department of Mathematical and Statistical Sciences
University of Alberta
Edmonton, Alberta T6G 2R3, Canada