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This answer addresses the second question: "If we assume the fibers of $\pi$ are curves of genus $0$, can we embed $X$ into $\mathbb{P}^2_S$?" The answer to this is also no (providing there is a singu …

Let $$X: \quad a_{0,0}x^2 + a_{1,1}y^2 + a_{2,2}z^2 + a_{0,1}xy + a_{0,2}xz + a_{1,2}yz = 0 \quad \subset \mathbb{P}^2 \times \mathbb{P}^5$$ be the total space of the family of all plane conics. Then …

If the Neron-Severi group of an abelian variety is $\mathbb{Z}$, then a principal polarisation, if it exists, is unique. This is the generic case, as well as for generic jacobians. To find counter-exa …

I will focus attention on smooth projective varieties $X$ over $k$ with $\mathrm{Pic}(X_{\bar{k}})$ a free finitely generated abelian group, as they illustrate all the essential behaviour relevant to …

I consider only smooth curves for simplicity. In which case this is expected to be true, but certainly not known in general. In fact, it is even expected that the Brauer-Manin obstruction is already e …

Yes this is true. It can be proved using the Hochschild-Serre spectral sequence plus Hilbert's theorem 90 (it is true more generally for any geometrically connected projective variety $X$). The Hochs …

Since nobody has answered this yet, here is my attempt, however Im not sure if it is exactly what you want. Hopefully it will at least lead you to a solution of your problem if you have not seen this …