Seminar Algebra and Geometry: Vladimir Tsygankov, Gutenberg Universität, Mainz)

The finite subgroups in the plane Cremona group over field of complex numbers

I will survey briefly known results about  the finite subgroups in the plane Cremona group over field of complex numbers. The emphasis will be done on the so called ''geometrical method''. By the method the study of the finite subgroups is reduced to  study of so called the ''minimal'' finite subgroups of automorphism groups of
nonsingular del Pezzo surfaces and conic bundles.

The description by equations of nonsingular Del Pezzo surfaces in weighted projective spaces and description of their automorphism groups is a very old completed problem. The description of the ''minimal'' finite subgroups of automorphism groups of nonsingular del Pezzo surfaces was done by I.V. Dolgachev and V.A. Iskovskikh in 2007. I constructed a method that allows us to describe  by equations in weighted projective spaces the nonsingular conic bundles with an action of ''minimal'' subgroup of automorphisms . I applied the method in case, when the ''minimal'' subgroup of automorphisms is nonsolvable.

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