Unboundedness of zero-cycles on higher dimensional Fano manifolds

TL;DR

This paper demonstrates that higher dimensional Fano manifolds generally lack boundedness properties for their zero-cycle groups, contrasting with del Pezzo surfaces, and shows that their ${ m CH}_0$ groups can be unbounded or infinite-dimensional.

Contribution

It provides the first evidence that higher dimensional Fano manifolds do not satisfy boundedness properties for their ${ m CH}_0$ groups, unlike lower-dimensional cases.

Findings

${ m CH}_0$ groups are unbounded for certain Fano hypersurfaces.

Quartic threefolds with odd degree points lack upper bounds on minimal odd degrees.

${ m CH}_0$ groups can be infinite-dimensional, indicating unboundedness.

Abstract

We show that, unlike del Pezzo surfaces, higher dimensional Fano manifolds do not satisfy in general boundedness properties for their ${\rm CH}_0$ group of $0$-cycles. For example, for quartic threefolds having a point of odd degree, there is no ``Coray type" uperbound on the minimal odd degrees of points. Also, the ${\rm CH}_0$-group of Fano hypersurfaces can be ``unbounded'' (a notion which is related to infinite dimensionality in the sense of Mumford), meaning that there is no integer $N$ such that $0$-cycles of degree at least $N$ are effective.