Relative Brauer groups of torsors of period two (Q288492)

The relative Brauer group \(\mathrm{Br}(Y/K)\) of a geometrically integral variety \(Y/K\) is the subgroup of \(\mathrm{Br}(K)\) consisting of classes which are split by the function field of \(Y\).NEWLINENEWLINEAmongst the main theorems are:NEWLINENEWLINETheorem 1.1. Let \(V\) be a genus one curve of period \(2\) and suppose \(X: y^2 = f(x)\) is a Weierstrass model for \(X = \mathrm{Jac}(V)\). Then there is an exact sequence NEWLINE\[NEWLINEX(K)/2 \to \mathrm{Br}(V/K) \to P(V)/I(V) \to 0.NEWLINE\]NEWLINENEWLINENEWLINETheorem 1.4. Let \(Y/K\) be a smooth projective and geometrically integral variety with Picard variety \(A = \mathrm{Pic}^0_Y\). Then there is an exact sequence NEWLINE\[NEWLINEA(K)/\mathrm{Pic}^0_Y \to \mathrm{Br}(Y/K) \to P(Y)/I(Y) \to 0.NEWLINE\]