SHLAFLY DOUBLE SIX

A double-six consists of a pair of ordered 6-tuples of lines (P_{0},…,P_{5}) and (Q_{0},…,Q_{5})
such that each P_{i} meets all the Q_{j} except Q_{i}.

For each i and j consider the intersection of the plane spanned by P_{i} and Q_{j} with that spanned by
P_{j} and Q_{i}; this gives a line R_{ij} which meets all the four lines. This gives us 6 + 6 + = 27
lines1 .

We will construct a Shlafly double-six and write the equation of the (unique) cubic that contains it.

Let F denote the smooth quadric in ℙ^{3} defined by XY - ZT = 0. This contains the lines:

- P
_{1}defined by X = Z = 0. - P
_{2}defined by Y = T = 0. - P
_{3}defined by X = T and Y = Z. - Q
_{4}defined by X = Z and Y = T. - Q
_{5}defined by Y = Z = 0. - Q
_{6}defined by X = T = 0.

This choice of lines means that P_{i} meets Q_{j} for i = 1,2,3 and j = 4,5,6. At the same time the P_{i}’s are
mutually disjoint; as are the Q_{j}’s. Conversely, given such a collection of six lines we can always choose
co-ordinates so that these lines are given by the equations above. As a consequence such a configuration
always lies in a (unique) smooth quadric.

We take a general linear polynomial aT + bX + cY + dZ and consider the cubic surface S defined by the equation

It is clear that the lines above are contained in S. Conversely, any cubic that contains the six lines defined above has this form.

We will construct the Shlafly double-six for S by suitably choosing the constants a, b, c, d, e. Now S also contains the lines:

- R
_{16}defined by X = 0 and aX + bY + cZ + dT = 0. - R
_{25}defined by Y = 0 and aX + bY + cZ + dT = 0. - R
_{34}defined by X + Y = Z + T and aX + bY + cZ + dT = 0.

We want a line Q_{1} which meets the lines P_{2}, P_{3} and R_{16}. Since the latter three are skew lines,
Q_{1} is uniquely determined by its intersection with P_{2} as follows. Let us pick a point x_{21} of
P_{2} given by (X : Y : Z : T) = (t : 0 : 1 : 0). Let π_{61} be the span of x_{21} and the line R_{16} and
let π_{31} be the span of x_{21} and the line P_{3}. Then Q_{1} is the line where the planes π_{61} and π_{31}
intersect.

In order that S contains Q_{1} we need to choose the constants a, b, c, d and e so that

If this is done then the cubic S contains Q_{1}. We claim that this cubic contains a Schlafly double-six which
we now construct.

We will repeatedly apply the following construction. Let L_{1}, L_{2}, M_{1} and M_{2} be lines in S such
that:

Let π be the plane containing L_{1} and L_{2}. The intersection of S and π contains L_{1} and L_{2}. The residual
intersection is then another line L_{3}. The intersection of M_{i} with the plane π does not lie on L_{1} or L_{2} and
hence must lie on on L_{3}. We can thus construct L_{3} as the line joining the intersection of M_{1} and π with the
intersection of M_{2} and π. We denote this line by I(L_{1},L_{2},M_{1},M_{2}).

The inductive construction of the remaining lines in the double-six is as follows: