Invariants and projections of six lines in projective space

Given six lines in P 3 \mathbf {P}^3 , quartics through the six lines define a map from P 3 \mathbf {P}^3 to P 4 \mathbf {P}^4 , and the image of this map is described in terms of invariants of the six lines. The map can be interpreted as projection of the six lines, and this permits a description of the canonical model of the octic surface which is given by points which project the lines so that they are tangent to a conic. We also define polarity for sets of six lines, and discuss the above map in the case of a self-polar set of lines and in the case of six lines which form a “double-sixer” on a cubic surface.