On the order three Brauer classes for cubic surfaces

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On the order three Brauer classes for cubic surfaces

3.3. Fact-Definition. –––– For every pair of Steiner trihedra, there are exactly

two other pairs having no line in common with the pair given. An ordered triple of

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

3.4. Remarks. –––– i) Together, 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 form

⎡ ⎤

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]), there 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, there 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

6

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