arXiv subjects
Publications and source records attributed to Sergey A. Gaifullin.
In 2007, Dubouloz introduced Danielewski varieties. Such varieties generalize Danielewski surfaces and provide counterexamples to generalized Zariski cancellation problem in arbitrary dimension. In the present work we describe the automorphism group of a Danielewski variety. This result is a generalization of a description of automorphisms of Danielewski surfaces due to Makar-Limanov.
In the end of the 19th century Bricard discovered a phenomenon of flexible polyhedra, that is, polyhedra with rigid faces and hinges at edges that admit non-trivial flexes. One of the most important results in this field is a theorem of Sabitov asserting that the volume of a flexible polyhedron is constant during the flexion. In this paper we study flexible polyhedral surfaces in the 3-space two-periodic with respect to translations by two non-colinear vectors that can vary continuously during the flexion. The main result is that the period lattice of a flexible two-periodic surface homeomorphic to a plane cannot have two degrees of freedom.
A description of transitive actions of a semisimple algebraic group G on toric varieties is obtained. Every toric variety admitting such an action lies between a product of punctured affine spaces and a product of projective spaces. The result is based on the Cox realization of a toric variety as a quotient space of an open subset of a vector space V by a quasitorus action and on investigation of the G-module structure of V.
We study the Cox realization of an affine variety, i.e., a canonical representation of a normal affine variety with finitely generated divisor class group as a quotient of a factorially graded affine variety by an action of the Neron-Severi quasitorus. The realization is described explicitly for the quotient space of a linear action of a finite group. A universal property of this realization is proved, and some results on the divisor theory of an abstract semigroup emerging in this context are given. We show that each automorphism of an affine variety can be lifted to an automorphism of the Cox ring normalizing the grading. It follows that the automorphism group of a non-degenerate affine toric variety of dimension $\geq 2$ has infinite dimension. We obtain a wild automorphism of the three-dimensional quadratic cone that rises to Anick's automorphism of the polynomial algebra in four variables.
In 1973 V.L.Popov classified affine SL(2)-embeddings. He proved that a locally transitive SL(2)-action on a normal affine three-dimensional variety X is uniquely determined by a pair (p/q, r), where 0 X is a dense open equivariant embedding. Then X is toric if and only if there exist a quasitorus T and a $(G\times T)$-module V such that $X\stackrel{G}{\cong} V//T$. The key role in the proof plays D. Cox's construction.