An interesting 0-cycle

Abstract

The geometric and arithmetic properties of a smooth algebraic variety $X$ are reflected by the configuration of its subvarieties. A principal invariant of these are the Chow groups $\CH^p(X)$, defined to be the group of codimension-$p$ algebraic cycles modulo rational equivalence. For $p=1$ these groups are classical and well understood. For $p\dgeqq 2$ they are nonclassical in character and constitute a major area of study. In particular, it is generally difficult to decide whether a given higher codimension cycle is or is not rationally equivalent to zero. In their study of the moduli spaces of algebraic curves, C. Faber and R. Pandharipande introduced a canonical $0$-cycle $z_K$ on the product $X=Y\times Y$ of a curve $Y$ with itself. This cycle is of degree zero and Albanese equivalent to zero, and they asked whether or not it is rationally equivalent to zero. This is trivially the case when the genus $g=0,1,2$, and they proved that this is true when $g=3$. It is also the case when $Y$ is hyperelliptic or, conjecturally, when it is defined over a number field. We show that $z_K$ is not rationally equivalent when $Y$ is general and $g\dgeqq 4$. The proof is variational, and for it we introduce a new computational method using Shiffer variations. The condition $g\dgeqq 4$ enters via the property that the tangent lines to the canonical curve at two general points must intersect.