6.1 A comment on Figure 1

It would be worthwhile for the reader to reflect on the relationship between the relations and the syzygies of $B_4$ and singularities of plane curves. One such codimension one singularity is the triple point , which corresponds to the last two relations above, which can be viewed as ``the motion of a double point across a line''. One such codimension two singularity is the quadruple point , and it corresponds to the syzygy of Figure 1: There is a circle-worth of generic deformations of the quadruple point, corresponding to ``the cross rotating around the target'': . The different codimension one singularities along this rotation are exactly the relations in our syzygy.