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.