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Splitting Brauer classes using the universal Albanese
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We prove that every Brauer class over a field splits over a torsor under an abelian variety. If the index of the class is not congruent to 2 modulo 4, we show that the Albanese variety of any smooth curve of positive genus that splits the class also splits the class, and there exist many such curves splitting the class. We show that this can be false when the index is congruent to 2 modulo 4, but adding a single genus 1 factor to the Albanese suffices to split the class.
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Cited by 1 Pith paper
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Some properties of a Brauer class
The Brauer class obstructing the relative Picard functor of a smooth proper curve carries an involution of the second kind and splits at generic points of theta divisors.
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