Unboundedness of zero-cycles on higher dimensional Fano manifolds

TL;DR

This work shows that higher-dimensional Fano manifolds generally lack boundedness for their CH_0 groups, in stark contrast with del Pezzo surfaces. It establishes unbounded CH_0 for very general even-degree hypersurfaces X ⊂ P^{n+1} (n ≥ 3) by Totaro-style specialization to characteristic 2, using the presence of nonzero differential forms on desingularizations to obstruct boundedness; it also demonstrates that quartic 3-folds cannot admit a Coray-type bound on odd-degree points. The approach blends Mumford–Roitman-type arguments with trace maps in inseparable extensions (through tensor rank), and leverages specialization techniques to transfer obstructions from special fibers to very general fibers. Additionally, the paper extends boundedness results to del Pezzo surfaces and provides a framework for understanding (un)boundedness of CH_0 across Fano varieties in characteristic 0 and beyond.

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.

Theorems & Definitions (61)

• Theorem 1.1
• Theorem 1.4
• Remark 1.5
• Definition 1.6
• Theorem 1.7
• Theorem 1.8
• Remark 1.9
• Theorem 1.10
• Theorem 1.11
• Theorem 1.12