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We prove an algebraic version of a classical theorem in topology, assertingthat an abelian p-group action on a smooth projective variety of positivedimension cannot fix exactly one point. When the group has only two elements,we prove that the number of fixed points cannot be odd. The main tool is aconstruction originally used by Rost in the context of the degree formula. Theframework of diagonalisable groups allows us to include the case of base fieldsof characteristic p.

Diagonalisable p-groups cannot fix exactly one point on projective varieties