arXiv subjects
Publications and source records attributed to S. A. Kudryavtsev.
The purely log terminal blow-ups of three-dimensional terminal toric singularities are described. The three-dimensional divisorial contractions $f\colon (Y,E)\to (X\ni P)$ are described provided that $\Exc f=E$ is an irreducible divisor, $(X\ni P)$ is a toric terminal singularity, $f(E)$ is a toric subvariety and $Y$ has canonical singularities.
The purely log terminal blow-ups of three-dimensional terminal toric singularities are described.
In this paper the three-dimensional divisorial contractions $f\colon Y\to (X\ni P)$ are classified provided that $\Exc f=E$ is an irreducible divisor, $f(E)=P$, the variety $Y$ has canonical singularities and $(X\ni P)$ is a toric terminal singularity.
In this paper the log surfaces without $\QQ$-complement are classified. In particular, they are non-rational always. This result takes off the restriction in the theory of complements and allows one to apply it in the most wide class of log surfaces.
All varieties, extremal contractions, singularities are divided on exceptional and non-exceptional ones. Roughly speaking, there are the infinite families of non-exceptional varieties, extremal contractions or singularities and only the finite number of the types of exceptional ones. This subdivision is well demonstrated by the example of Du Val singularities. There are two infinite series of non-exceptional singularities: $A_n$ and $D_n$ and only three types of exceptional ones: $E_6$, $E_7$ and $E_8$. Also the importance of exceptionality phenomenon follows from the next observation: A). If a variety, extremal contraction or singularity is non-exceptional then the linear system $|-nK_X|$ must have a "good" member for small $n$. For example we can take $n\in \{1,2\}$ for the two-dimensional singularities and $n\in \{1,2,3,4,6\}$ for the three-dimensional singularities. B). Exceptional ones are "bounded" and can be classified. Using the inductive method of algebraic variety classification it was obtained the description of three-dimensional exceptional hypersurface singularities in this paper.
The first examples of exceptional terminal singularities are constructed.
In this paper we prove the existence of purely log terminal blow-up for Kawamata log terminal singularity and obtain the criterion for a singularity to be weakly exceptional in terms of the exceptional divisor of plt blow-up.