The minimal dimension of a faithful linear representation of all finite subgroups of the two-dimensional Cremona group is 8 when the field contains a primitive cube root of unity, 6 otherwise, and infinite in positive characteristic.
Automorphisms of cubic surfaces in positive characteristic
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abstract
We classify all possible automorphism groups of smooth cubic surfaces over an algebraically closed field of arbitrary characteristic. As an intermediate step we also classify automorphism groups of quartic del Pezzo surfaces. We show that the moduli space of smooth cubic surfaces is rational in every characteristic, determine the dimensions of the strata admitting each possible isomorphism class of automorphism group, and find explicit normal forms in each case. Finally, we completely characterize when a smooth cubic surface in positive characteristic, together with a group action, can be lifted to characteristic zero.
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Representations of finite subgroups of Cremona groups
The minimal dimension of a faithful linear representation of all finite subgroups of the two-dimensional Cremona group is 8 when the field contains a primitive cube root of unity, 6 otherwise, and infinite in positive characteristic.