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Topic

Torsion in Chow Groups of Quadrics

Department of Mathematics and Statistics

Date & location

• Tuesday, December 16, 2025
• 2:00 P.M.
• David Turpin Building, Room A203

Examining Committee

Supervisory Committee

• Dr. Stephen Scully, Department of Mathematics and Statistics, ӣ��Ӱ�� (Supervisor)
• Dr. Heath Emerson, Department of Mathematics and Statistics, UVic (Member)

External Examiner

• Dr. Stefan Gille, Department of Mathematical and Statistical Sciences, University of Alberta

Chair of Oral Examination

• Dr. Leslee Francis Pelton, Department of Curriculum and Instruction, UVic

Abstract

We investigate here the Chow groups of arbitrary projective quadrics over a field 𝑘 of characteristic 2. Much work has been done on the Chow groups of quadrics over fields of characteristic different from 2, but relatively little is known when the characteristic is 2, and even less so when the quadrics are allowed to be singular. For a smooth variety, the codimension one Chow group CH¹(𝑋) is naturally isomorphic to the Picard group Pic(𝑋), but this is not necessarily true when 𝑋 is singular, even if 𝑋 is a quadric. We therefore begin by computing Pic(𝑋) for an arbitrary anisotropic quadric 𝑋. This allows us to generalize the well-known result stating that two nonsingular quadratic forms are similar if and only if their corresponding quadrics are isomorphic to the case of anisotropic singular forms.

One important case in characteristic not 2 in which CH(𝑋) is fully understood is the case where 𝑋 is an excellent quadric. This computation (due to Rost, Karpenko and Merkurjev) rests on two key ingredients: (i) Rost’s determination of the Chow groups of affine norm quadrics, and (ii) the action of cohomological Steenrod operations on the mod-2 Chow groups of smooth varieties. For many years, (ii) was only available over fields of characteristic different from 2, but recent work of Primozic has removed the characteristic requirement. In the present work, we extend (i) to the characteristic-2 case, thereby allowing us to determine the Chow groups of all excellent quadrics in this setting. We then begin a systematic study of the torsion subgroup 𝑇CH^𝑝(𝑋) ⊂ CH^𝑝(𝑋) for an arbitrary (possibly singular) quadric 𝑋. We completely resolve the 𝑝 = 1 case, and provide strong partial results for the 𝑝 = 2 case.