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.

About the authors

Igor Vladimirovich Dolgachev

University of Michigan, Department of Mathematics

Email: idolga@umich.edu
PhD

Alexander Duncan

University of South Carolina

Email: duncan@math.sc.edu
PhD

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