brauer groups, tamagawa measures, and rational points

16 THE BRAUER-SEVERI VARIETY OF A CENTRAL SIMPLE ALGEBRA [Chap. I
i : Y ֒→ P N L . By Sublemma 3.11, there exists a hypersurface H y such that
H y ⊃ i(Y ) \ i(Y ) and i(σ(y)) ∉ H y for every σ ∈ G. Here, i(Y ) denotes the
closure of i(Y ) in P N L . By construction, the morphism
i| Y \i −1 (H y ) : Y \ i −1 (H y ) −→ P N L \ H y
is a closed embedding. As P N L \ H y is an affine scheme, Y \ i −1 (H y ) must be
affine, too. Hence,
O y := \
σ −1 (Y \ i −1 (H y )) ⊂ Y
σ∈G
is the intersection of finitely many affine open subsets in a quasi-projective and,
therefore, separated scheme. Thus, O y is an affine open subset. By construction,
O y is G-invariant and contains y.
□
3.11. Sublemma. –––– Let L be a field and Z P N L be a Zariski-closed subset.
Further, let p 1 , . . . , p n ∈ P N L be finitely many closed points not contained in Z.
Then, there exists a hypersurface H ⊂ P N L which contains Z but does not contain
any of the points p 1 , . . . , p n .
Proof. We will give two proofs, an elementary one and a more canonical one
which uses cohomology of coherent sheaves.
First proof. Let S := L[X 0 , . . . , X N ] be the homogeneous coordinate ring for
the projective space P N L . S is a graded L-algebra. For d ∈Æ, we will denote by
S d the L-vector space of homogeneous elements of degree d.
We proceed by induction. The case n = 0 is trivial. Thus, assume that the
assertion is true for n − 1 and consider n points p 1 , . . . , p n . By induction
hypothesis, there exists a homogeneous element s ∈ S, i.e., a hypersurface
H := V (s) of a certain degree d, such that H ⊇ Z and p 1 , . . . , p n−1 ∉ H .
We may assume p n ∈ H as, otherwise, the proof would be complete.
Z ∪ {p 1 } ∪ . . . ∪ {p n−1 } is a Zariski closed subset of P N L not containing p n.
Therefore, there exist some d ′ ∈Æand some homogeneous s ′ ∈ S d ′ such that
V (s ′ ) ⊇ Z ∪ {p 1 } ∪ . . . ∪ {p n−1 } but p n ∉ V (s ′ ). Every hypersurface
V (a · s ′d + b · s d ′ ) for a, b ∈ L non-zero contains Z but neither p 1 , . . . , p n−1 ,
nor p n .
Second proof. Tensoring the canonical exact sequence
0 −→ I {p1 , ... ,p n } −→ O X −→ O {p1 , ... ,p n } −→ 0
with the ideal sheaf I Z yields an exact sequence
0 −→ I {p1 , ... ,p n }∪Z −→ I Z −→ O {p1 , ... ,p n } −→ 0