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Title:
Birational maps of Severi-Brauer surfaces, with applications to Cremona groups of higher rank
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Abstract:
We describe the group of birational transformations of a non-trivial Severi-Brauer surface over a perfect field, proving that if it contains a point of degree 6, then it is not generated by elements of finite order. We then use this result to study Mori fibre spaces over the field of complex numbers and deduce that any group of cardinality at most C (the field of complex numbers) is a quotient of the Cremona group of rank at least 4. Moreover, we prove that the 3-torsion in the abelianization of the Cremona group of rank at least 4 is uncountable. This is based on a joint work with J. Blanc and E. Yasinsky.
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