On the order three Brauer classes for cubic surfaces

3.3. Fact-Definition. –––– For every pair of Steiner trihedra, **the**re are exactly

two o**the**r pairs having no line in common with **the** pair given. An **order**ed triple of

pairs of Steiner trihedra obtained in this way will be called a triplet.

3.4. Remarks. –––– i) Toge**the**r, a triplet contains all **the** 27 lines.

ii) There are 240 triplets on a non-singular **cubic** surface, corresponding to **the** 40 decompositions

of **the** 27 lines into **three** pairs of Steiner trihedra. Clearly, **the** operation

of W(E 6 ) is transitive on triplets and, thus, on decompositions.

The largest subgroup U t of W(E 6 ), stabilizing a triplet, is of **order** 216. It is a subgroup

of index two in [(S 3 ×S 3 )⋊/2]×S 3 . As an abstract group, U t

∼ = S3 ×S 3 ×S 3 .

3.5. Notation. –––– Let l 1 , . . .,l 9 be **the** nine lines defined by a Steiner trihedron.

Then, we will denote **the** corresponding pair of Steiner trihedra by a rectangular

symbol of **the** **for**m

⎡ ⎤

l 1 l 2 l 3

⎣l 4 l 5 l 6

⎦.

l 7 l 8 l 9

The planes of **the** trihedra contain **the** lines noticed in **the** rows and columns.

3.6. –––– To describe Steiner trihedra explicitly, one works best in **the** blownup

model. In Schläfli’s notation [Sch, p.116] (cf. Hartshorne’s notation in [Ha,

Theorem V.4.9]), **the**re are 20 pairs of type I,

⎡ ⎤

a i b j c ij

⎣ b k c jk a j

⎦,

c ik a k b i

10 pairs of type II, ⎡ ⎤

c il c jm c kn

⎣ c jn c kl c im

⎦,

c km c in c jl

and 90 pairs of type III,

⎡ ⎤

a i b l c il

⎣ b k a j c jk

⎦.

c ik c jl c mn

3.7. –––– Consequently, **the**re are two types of decompositions of **the** 27 lines,

10 consist of two pairs of type I and one pair of type II,

⎧⎡

⎤ ⎡ ⎤ ⎡ ⎤⎫

⎨ a i b j c ij a l b m c lm c il c jm c kn ⎬

⎣ b k c jk a j

⎦, ⎣ b n c mn a m

⎦, ⎣ c jn c kl c im

⎦

⎩

⎭ ,

c ik a k b i c ln a n b l c km c in c jl

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