On the order three Brauer classes for cubic surfaces

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On the order three Brauer classes for cubic surfaces

is bijective.

Proof. This proof has a computer part. Using gap, we made the observation that

rk Pic(S ) H = 1 for every H ⊂ W(E 6 ) such that H 1 (H, Pic(S )) ∼ =/3.

To verify the assertion, we will show injectivity. According to Lemma 3.17.i), it

suffices to test this on the 3-Sylow subgroups of H and U t . But the 3-Sylow subgroup

∼ =/3×/3×/3is abelian. Thus, Lemma 3.17.ii) immediately implies

the assertion.


U (3)

t

3.19. Corollary. –––– Let H ′ ⊆ H ⊆ U t be arbitrary. Then, for the restriction

map res: H 1 (H, Pic(S )) −→ H 1 (H ′ , Pic(S )), there are the following limitations.

i) If H 1 (H, Pic(S )) = 0 then H 1 (H ′ , Pic(S )) = 0.

ii) If H 1 (H, Pic(S )) ∼ =/3and H 1 (H ′ , Pic(S )) ≠ 0 then res is an injection.

iii) If H 1 (H, Pic(S )) ∼ =/3×/3then H 1 (H ′ , Pic(S )) ∼ =/3×/3or 0.

In the former case, H ′ = H. In the latter case, H ′ = 0.

Proof. We know from Remark 3.16.i) that both groups may be only 0,/3, or

/3×/3.

i) If H 1 (H ′ , Pic(S )) were isomorphic to/3or/3×/3then the restriction

from U t to H ′ would be the zero map.

ii) is immediate from the computations above.

iii) This assertion is obvious as H is of order three.


4 Computing the Brauer-Manin obstruction

A splitting field for the Brauer class.

4.1. –––– The theory developed above shows that the non-trivial Brauer classes

are, in a certain sense, always the same, as long as they are of order three. This may

certainly be used for explicit computations. The Brauer class remains unchanged

under suitable restriction maps. These correspond to extensions of the base field.

We will, however, present a different method here. Its advantage is that it avoids

large base fields.

4.2. Lemma. –––– Let S be a non-singular cubic surface and let T 1 and T 2 be

two pairs of Steiner trihedra that define disjoint sets of lines.

Then, the 18 lines defined by T 1 and T 2 contain exactly three double-sixes.

These form a triple of azygetic double-sixes.

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