New perspectives for canonical dimension
Final Report Abstract
The main findings of this project belong to the first direction mentioned in the proposal’s summary, namely using in new contexts methods originally devised for the study of rational points on projective homogeneous varieties. In particular techniques traditionally used for applications of arithmetic nature (such as the degree formula) have been successfully used in the purely geometric context of symmetries of algebraic varieties (fixed points of group actions) within this project. Substantial progress was achieved on the first explicit problem mentioned in the proposal’s summary, a question of Serre (asking whether a p-group acting on the affine space over a field of characteristic not p must fix a rational point). Several new problems concerning finite group actions on varieties emerged, inspired by the work of algebraic topologists in the sixties, and some of them have been solved under the funding received for this project. One particular result obtained asserts that symmetries of (positive dimensional, smooth projective) algebraic varieties whose order is a prime power can never have exactly one fixed point: there is either none, or at least two. One can say more concerning symmetries of order two, that is, involutions: the number of fixed points must be even or infinite.
Publications
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On p-group actions on smooth projective varieties. Oberwolfach Reports, 13 (2016) no. 1, 228–230
Olivier Haution
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On rational fixed points of finite group actions on the affine space, Transactions of the American Mathematical Society, 369 (2017), 8277–8290
Haution, Olivier
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Diagonalisable p-groups acting on projective varieties cannot fix exactly one point, Journal of Algebraic Geometry
Haution, Olivier
