48 research outputs found
Birationally rigid Fano hypersurfaces with isolated singularities
It is proved that a general Fano hypersurface of index 1 (in the projective
space) with isolated singularities of general position is birationally rigid.
Therefore it cannot be fibered into uniruled varieties of a smaller dimension
by a rational map. The group of birational self-maps is either trivial or
cyclic of order two.Comment: 21 pages, LATe
Birational geometry of Fano direct products
We prove birational superrigidity of direct products of primitive Fano varieties of the following two types: either
is a general hypersurface of degree , ,
or is a general double space of index
1, . In particular, each structure of a rationally connected fiber
space on is given by a projection onto a direct factor. The proof is based
on the connectedness principle of Shokurov and Koll\' ar and the technique of
hypertangent divisors.Comment: 38 pages, LaTeX. This is the final enlarged version of the pape
Birationally rigid Fano hypersurfaces
We prove that a smooth Fano hypersurface , ,
is birationally superrigid. In particular, it cannot be fibered into uniruled
varieties by a non-trivial rational map and each birational map onto a minimal
Fano variety of the same dimension is a biregular isomorphism. The proof is
based on the method of maximal singularities combined with the connectedness
principle of Shokurov and Koll\' ar.Comment: 31 pages, LATe
Birational geometry of algebraic varieties, fibred into Fano double spaces
We develop the quadratic technique of proving birational rigidity of
Fano-Mori fibre spaces over a higher-dimensional base. As an application, we
prove birational rigidity of generic fibrations into Fano double spaces of
dimension and index one over a rationally connected base of
dimension at most . An estimate for the codimension of the
subset of hypersurfaces of a given degree in the projective space with a
positive-dimensional singular set is obtained, which is close to the optimal
one.Comment: 30 pages, the final versio
The -inequality for complete intersection singularities
The famous -inequality is extended to generic complete intersection
singularities: it is shown that the multiplicity of the self-intersection of a
mobile linear system with a maximal singularity is higher than , where
is the multiplicity of the singular point.Comment: 9 pages, the final versio
Birational geometry of algebraic varieties with a pencil of Fano complete intersections
We prove birational superrigidity of generic Fano fiber spaces , the fibers of which are Fano complete intersections of index 1 and
dimension in , provided that . The proof
combines the traditional quadratic techniques of the method of maximal
singularities with the linear techniques based on the connectedness principle
of Shokurov and Koll\' ar. Certain related results are also considered.Comment: 15 pages, LATE