Every generically trivial torsor under a smooth group scheme over a smooth variety over any field is Zariski semilocally trivial, settling the Grothendieck-Serre question.
Realisation of linear algebraic groups as automorphism groups
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abstract
Let $G$ be a linear algebraic group, over a field $F$. We show that $G$ is isomorphic to the automorphism group scheme of a smooth projective $F$-variety, defined as the blow-up of a projective space, along a suitable smooth subvariety.
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Generically trivial torsors under constant groups
Every generically trivial torsor under a smooth group scheme over a smooth variety over any field is Zariski semilocally trivial, settling the Grothendieck-Serre question.