Finiteness theorems for algebraic cycles of small codimension on quadric fibrations over curves

TL;DR

This paper proves finiteness results for algebraic cycles of small codimension on quadric fibrations over curves, especially over finitely generated fields, with specific conditions on the relative dimension.

Contribution

It establishes new finiteness theorems for algebraic cycles on quadric fibrations over curves, extending understanding in algebraic geometry.

Findings

CH^{i}(X) is finitely generated for i<=4 when the base field is finitely generated over Q and the fibration has odd relative dimension at least 11.

Finiteness theorems are obtained for algebraic cycles of small codimension on quadric fibrations over perfect fields.

Results apply to cases where the base field is finitely generated over Q and the relative dimension meets certain criteria.

Abstract

We obtain finiteness theorems for algebraic cycles of small codimension on quadric fibrations X over curves over perfect fields k. For example, if k is finitely generated over Q and the fibration has odd relative dimension at least 11, then CH^{i}(X) is finitely generated for i<=4.