Prelog Chow rings and degenerations

TL;DR

This paper introduces prelog Chow rings and groups for simple normal crossing varieties, extending degeneration methods to study cycle classes and stable irrationality in degenerations, including toric cases.

Contribution

It defines and explores properties of prelog Chow rings and groups, enabling the use of degeneration techniques in more singular and complex degenerations.

Findings

Prelog Chow rings extend cycle class specialization in degenerations.

Application to degenerations of cubic surfaces.

Illustration with elliptic curve degenerations.

Abstract

For a simple normal crossing variety $X$, we introduce the concepts of prelog Chow ring, saturated prelog Chow group, as well as their counterparts for numerical equivalence. Thinking of $X$ as the central fibre in a (strictly) semistable degeneration, these objects can intuitively be thought of as consisting of cycle classes on $X$ for which some initial obstruction to arise as specializations of cycle classes on the generic fibre is absent. Cycle classes in the generic fibre specialize to their prelog counterparts in the central fibre, thus extending to Chow rings the method of studying smooth varieties via strictly semistable degenerations. After proving basic properties for prelog Chow rings and groups, we explain how they can be used in an envisaged further development of the degeneration method by Voisin et al. to prove stable irrationality of very general fibres of certain families of varieties; this extension would allow for much more singular degenerations, such as toric degenerations as occur in the Gross-Siebert programme, to be usable. We illustrate that by looking at the example of degenerations of elliptic curves, which, although simple, shows that our notion of prelog decomposition of the diagonal can also be used as an obstruction in cases where all components in a degeneration and their mutual intersections are rational. We also compute the saturated prelog Chow group of degenerations of cubic surfaces.