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Degree and the Brauer-Manin obstruction. (English) Zbl 1408.14086
Let $$X$$ be a smooth, projective, geometrically integral variety over a number field $$k$$. It is well-known that in general the Hasse principle may fail for $$X$$ so that $$X(k)$$ is empty while $$X(\mathbb{A}_k)$$ is not. Sometimes (but not always – see [A. N. Skorobogatov, Invent. Math. 135, No. 2, 399–424 (1999; Zbl 0951.14013)] for a bielliptic counterexample) this failure can be explained by the Brauer-Manin obstruction. More precisely, there is a pairing $X(\mathbb{A}_k)\times \mathrm{Br}(X)\to \mathbb{Q}/\mathbb{Z}$ between the adelic points of $$X$$ and its Brauer group such that for any subgroup $$B\subset\mathrm{Br}(X)$$, the rational points $$X(k)$$ are contained in the set $$X(\mathbb{A}_k)^B$$ of adelic points which are orthogonal to all elements of $$B$$.
Recent literature deals with the question how the torsion order of a Brauer class relates to the obstruction it can give. For a positive integer $$d$$, denote by $$B[d^\infty]$$ the $$d$$-primary part of $$B$$ and by $$B[d^\perp]$$ the prime-to-$$d$$ part.
Say that the property $$\mathrm{BM}_{B,d}$$ is satisfied for $$X$$ if $$B[d^\infty]$$ captures any Brauer-Manin obstruction to the Hasse principle, i.e. the following implication holds: $X(\mathbb{A}_k)^B=\emptyset \Rightarrow X(\mathbb{A}_k)^{B[d^\infty]}=\emptyset.$ Say that $$\mathrm{BM}_{B,d}^\perp$$ is satisfied for $$X$$ if $$B[d^\perp]$$ never obstructs the Hasse principle, i.e. if the following implication holds: $X(\mathbb{A}_k)\neq\emptyset \Rightarrow X(\mathbb{A}_k)^{B[d^\perp]}\neq\emptyset.$ We omit the index $$B$$ in the notation when $$B=\mathrm{Br}(X)$$.
It has been shown that $$\mathrm{BM}_2^\perp$$ holds for diagonal quartic surfaces over $$\mathbb{Q}$$ [E. Ieronymou and A. N. Skorobogatov, Adv. Math. 270, 181–205 (2015; Zbl 1388.14071), E. Ieronymou and A. N. Skorobogatov, Adv. Math. 307, 1372–1377 (2017; Zbl 1388.14072)] (with a generalisation to number fields in [M. Nakahara, Adv. Math. 348, 512–522 (2019)]), Kummer varieties [A. N. Skorobogatov and Y. G. Zarhin, Pure Appl. Math. Q. 13, No. 2, 337–368 (2017; Zbl 1398.14029)] and del Pezzo surfaces of degree $$2$$ [M. Nakahara, Adv. Math. 348, 512–522 (2019)]. On the other hand, in [P. Corn and M. Nakahara, Res. Number Theory 4, No. 3, Art. 33 (2018)] resp. [J. Berg and A. Várilly-Alvarado, “Odd order obstructions to the Hasse principle on general K3 surfaces”, arXiv:1808.00879] an obstruction by an algebraic resp. transcendental $$3$$-torsion Brauer class on a degree $$2$$ K3 surface was constructed.
The present article relates the above properties to the degree of a variety. The authors ask whether degrees capture the Brauer-Manin obstruction on $$X$$, which is defined to mean that $$\mathrm{BM}_d$$ holds for any $$d$$ which is the degree of a globally generated ample line bundle on $$X$$. Apart from being a natural question, an answer may be relevant for algorithmic purposes.
The first main result is that torsors under abelian varieties satisfy $$\mathrm{BM}_{B,d}$$ for any subgroup $$B$$ and any $$d$$ divisible by the period of the variety, in particular for any $$d$$ which is the degree of a globally generated ample line bundle on $$X$$. Taking $$B=\mathrm{Br}(X)[d^\perp]$$, this shows that any prime-to-$$d$$ Brauer class cannot obstruct. The proof relies on the observation that for any class $$\mathcal{A}$$ in $$B[d^\perp]$$, there exists an étale automorphism of $$X$$ which annihilates $$\mathcal{A}$$. For $$B=\mathrm{Br}(X)$$ and semi-abelian varieties, one could also prove the theorem using Manin’s work under the assumption of finiteness of the Tate-Shafarevich group.
The authors then proceed to consider quotients of torsors under abelian varieties and prove that Kummer varieties satisfy $$\mathrm{BM}_2$$. In combination with the aforementioned result by Skorobogatov-Zarhin, the appendix strengthens this to $$\mathrm{BM}_{B,2}$$ for any subgroup $$B$$. It is also shown that assuming the finiteness of the Tate-Shafarevich group, degrees capture the Brauer-Manin obstruction on bielliptic surfaces.
The last section proves that degrees capture the Brauer-Manin obstruction on minimal rational conic bundles and Severi-Brauer bundles over elliptic curves with finite Tate-Shafarevich group, which combined with previous results on del Pezzo surfaces implies an affirmative answer for all geometrically rational minimal surfaces. Finally a counterexample of a conic bundle over an elliptic curve, where degrees do not capture the Brauer-Manin obstruction, is given.
##### MSC:
 14G05 Rational points 11G35 Varieties over global fields 14F22 Brauer groups of schemes
##### Keywords:
Brauer-Manin obstruction; degree; period; rational points
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