in joint work with g. blekherman we proved that a truncated multisequence $y$ of degree $m$ has a representing measure in the truncated moment problem if and only if its core variety $v(y)$ is nonempty, in which case $v(y)$ coincides with the union of the supports of all representing measures. in general, for a given numerical sequence $y$, it may be quite difficult to compute $v(y)$ or even to determine if it is nonempty. we illustrate some cases where we can compute $v(y)$ or can otherwise describe it concretely.