Given an algebraic surface X, we can define the Chow group of points on X: take the abelian group generated by points of X, and declare an element to be 0 if it represents the zeros and poles of a rational function on some curve lying in X. These Chow groups are in general notoriously difficult to compute. Bloch and Beilinson made a number of far-reaching conjectures about the structure of Chow groups, but to date there is very little evidence for them. I will not spend much time discussing the conjectures themselves, but one concrete consequence considers a particular type of surface X, and claims that certain formal sums of points on X should be zero in Chow. We will verify that this holds for a few specific examples, by searching for explicit curves in X that exhibit the desired relations.