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Automorphisms of affine varieties

Abstract : The thesis consists of two parts. The first part is dedicated to transformations of finite-dimensional algebras. It is easy to see that the automorphism group of a finite-dimensional algebra is an affine algebraic group. N.L.~Gordeev and V.L.~Popov proved that any affine algebraic group is isomorphic to the automorphism group of some finite-dimensional algebra. We use a similar approach to prove that any affine algebraic monoid can be obtained as the endomorphisms' monoid of a finite-dimensional algebra. Next, we study the solvability of automorphism groups of commutative Artin algebras. We introduce a criterion of their solvability and apply it to complete intersections and to isolated hypersurface singularities. We also study extremal cases of the introduced criterion. The second part of the thesis is dedicated to the infinite transitivity of special automorphism groups of affine and quasiaffine varieties. This property is equivalent to the flexibility for affine varieties. Firstly, we prove the equivalence of transitivity and infinite transitivity of special automorphism groups over algebraically closed field of arbitrary characteristic. Then we provide the criterion of flexibility for affine cones over projective varieties and apply it to del Pezzo surfaces of degree 4 and 5. Finally, we study flexibility of universal torsors over varieties covered by affine spaces and provide a wide range of families of flexible varieties.
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Aleksandr Perepechko. Automorphisms of affine varieties. General Mathematics [math.GM]. Université de Grenoble; Institut stran Azii i Afriki (Moskva), 2013. English. ⟨NNT : 2013GRENM065⟩. ⟨tel-01168473⟩



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