Proof of Prop. 10.1

Here we verify the claim in the proof of Proposition 10.1 that, for each polygon on the cubic surface, there exist six pairwise disjoint lines that are disjoint from the polygon.
The following list of polygons are taken from the Section 3.

truePolygons=Sort[{{"E4","F24","G2"},{"E4","F34","G3"},{"E5","F15","G1"},{"E5","F25","G2"},{"E6","F16","G1"},{"E6","F36","G3"},{"F14","F25","F36"},{"F14","F26","F35"},{"F15","F26","F34"},{"F16","F24","F35"},{"E1","E5","G3","G4"},{"E1","E5","G3","G6"},{"E1","E6","F16","G5"},{"E1","E6","G2","G4"},{"E1","E6","G2","G5"},{"E1","E6","G3","G4"},{"E1","F12","F14","F36"},{"E1","F12","F14","G1"},{"E1","F12","F15","F36"},{"E1","F13","F14","G1"},{"E1","F13","F16","F25"},{"E1","F15","F36","G6"},{"E1","F16","F25","G5"},{"E2","E4","G1","G5"},{"E2","E4","G1","G6"},{"E2","E4","G3","G6"},{"E2","E5","F25","G4"},{"E2","E5","G1","G6"},{"E2","E5","G3","G4"},{"E2","E5","G3","G6"},{"E2","F12","F25","F34"},{"E2","F12","F26","F34"},{"E2","F12","F26","G2"},{"E2","F15","F23","F24"},{"E2","F15","F24","G5"},{"E2","F23","F26","G2"},{"E2","F25","F34","G4"},{"E3","E4","G1","G5"},{"E3","E4","G1","G6"},{"E3","E4","G2","G5"},{"E3","E6","G2","G4"},{"E3","E6","G2","G5"},{"E3","F13","F24","F36"},{"E3","F13","F35","G3"},{"E3","F16","F23","F34"},{"E3","F16","F23","F35"},{"E3","F16","F34","G6"},{"E3","F23","F35","G3"},{"E3","F24","F36","G4"},{"E4","F14","F23","F45"},{"E4","F14","F23","F46"},{"E4","F14","F45","G4"},{"E4","F14","F46","G4"},{"E4","F24","F45","G4"},{"E4","F34","F46","G4"},{"E5","F12","F35","F45"},{"E5","F12","F35","F56"},{"E5","F12","F45","G2"},{"E5","F15","F56","G5"},{"E5","F35","F45","G5"},{"E5","F35","F56","G5"},{"E6","F13","F26","F46"},{"E6","F13","F26","F56"},{"E6","F13","F56","G1"},{"E6","F26","F46","G6"},{"E6","F26","F56","G6"},{"E6","F36","F46","G6"},{"F12","F14","F56","G1"},{"F12","F15","F36","F46"},{"F12","F25","F34","F46"},{"F12","F26","F35","F45"},{"F12","F26","F45","G2"},{"F13","F14","F26","F56"},{"F13","F14","F56","G1"},{"F13","F16","F24","F45"},{"F13","F16","F25","F45"},{"F13","F24","F36","F45"},{"F13","F25","F36","F45"},{"F13","F35","F46","G3"},{"F14","F23","F35","F46"},{"F14","F25","F46","G4"},{"F15","F23","F24","F56"},{"F15","F23","F34","F56"},{"F15","F24","F56","G5"},{"F15","F26","F46","G6"},{"F15","F36","F46","G6"},{"F16","F23","F34","F56"},{"F16","F25","F45","G5"},{"F16","F34","F56","G6"},{"F23","F26","F45","G2"},{"F23","F35","F46","G3"},{"F24","F35","F56","G5"},{"F24","F36","F45","G4"},{"F25","F34","F46","G4"},{"E1","E5","F12","F15","G1"},{"E1","E5","F15","G1","G6"},{"E1","E5","F25","G2","G4"},{"E1","F25","G2","G5"},{"E1","E6","F13","F16","G1"},{"E1","F36","G3","G6"},{"E1","F13","F14","F25","F36"},{"E2","E4","F23","F24","G2"},{"E2","E4","F24","G2","G5"},{"E2","F34","G3","G4"},{"E2","E5","F12","F25","G2"},{"E2","F15","G1","G5"},{"E2","F15","F23","F26","F34"},{"E3","E4","F23","F34","G3"},{"E3","F24","G2","G4"},{"E3","E4","F34","G3","G6"},{"E3","E6","F13","F36","G3"},{"E3","E6","F16","G1","G5"},{"E3","F16","G1","G6"},{"E3","E6","F36","G3","G4"},{"E3","F13","F16","F24","F35"},{"E4","F23","F24","F45","G2"},{"E4","F23","F34","F46","G3"},{"E5","F12","F15","F56","G1"},{"E5","F25","F45","G2","G5"},{"E6","F13","F36","F46","G3"},{"E6","F16","F56","G1","G6"},{"F12","F14","F25","F36","F46"},{"F12","F14","F26","F35","F56"},{"F12","F15","F26","F34","F46"},{"F13","F14","F26","F35","F46"},{"F14","F23","F26","F35","F45"},{"F14","F25","F36","F45","G4"},{"F15","F26","F34","F56","G6"},{"F16","F23","F24","F35","F56"},{"F16","F24","F35","F45","G5"}}];

We generate 30 groups of six pairwise disjoint lines and verify that for each polygon from the list above, one of these 30 groups works.

disjointSix={{"F23","F24","F25","F26","E1","G1"},{"F23","F34","F35","F36","E1","G1"},{"F24","F34","F45","F46","E1","G1"},{"F25","F35","F45","F56","E1","G1"},{"F26","F36","F46","F56","E1","G1"},{"F13","F14","F15","F16","E2","G2"},{"F13","F34","F35","F36","E2","G2"},{"F14","F34","F45","F46","E2","G2"},{"F15","F35","F45","F56","E2","G2"},{"F16","F36","F46","F56","E2","G2"},{"F12","F14","F15","F16","E3","G3"},{"F12","F24","F25","F26","E3","G3"},{"F14","F24","F45","F46","E3","G3"},{"F15","F25","F45","F56","E3","G3"},{"F16","F26","F46","F56","E3","G3"},{"F12","F13","F15","F16","E4","G4"},{"F12","F23","F25","F26","E4","G4"},{"F13","F23","F35","F36","E4","G4"},{"F15","F25","F35","F56","E4","G4"},{"F16","F26","F36","F56","E4","G4"},{"F12","F13","F14","F16","E5","G5"},{"F12","F23","F24","F26","E5","G5"},{"F13","F23","F34","F36","E5","G5"},{"F14","F24","F34","F46","E5","G5"},{"F16","F26","F36","F46","E5","G5"},{"F12","F13","F14","F15","E6","G6"},{"F12","F23","F24","F25","E6","G6"},{"F13","F23","F34","F35","E6","G6"},{"F14","F24","F34","F45","E6","G6"},{"F15","F25","F35","F45","E6","G6"}};

In the output, the first set is the polygon and the second is the six pairwise disjoint lines.

For[i=1,i<131,i++,
j=1;
While[Not[Intersection[truePolygons[[i]],disjointSix[[j]]]==={}],j++
];
Print["Polygon=",truePolygons[[i]]," Lines=",disjointSix[[j]]]
]