27 research outputs found
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
Birationally Rigid Finite Covers of the Projective Space
In this paper we prove birational superrigidity of finite covers of degree
of the -dimensional projective space of index 1, where
and , with at most quadratic singularities of rank , satisfying certain regularity conditions. Up to now, only cyclic covers
were studied in this respect. The set of varieties with worse singularities or
not satisfying the regularity conditions is of codimension
in the natural parameter space of the family.Comment: 16 page
Factorial hypersurfaces
In this paper the codimension of the complement to the set of factorial hypersurfaces of degree in is estimated for ,
Remarks on Galois Rational Coverings
In this note we improve the theorem on Galois rational covers
for primitive Fano varieties , recently proven by the
author, in the two directions: we extend to the maximum the class of Galois
groups , for which the proof works, and relax the conditions that must be
satisfied by the variety -- the divisorial canonicity alone is sufficient.Comment: 9 page
Birational geometry of singular Fano hypersurfaces of index two
For a Zariski general (regular) hypersurface V of degree M in the
Birationally rigid complete intersections of high codimension
We prove that a Fano complete intersection of codimension and index 1 in
the complex projective space for and
with at most multi-quadratic singularities is
birationally superrigid. The codimension of the complement to the set of
birationally superrigid complete intersections in the natural parameter space
is shown to be at least . The proof is based on the
techniques of hypertangent divisors combined with the recently discovered
-inequality for complete intersection singularities.Comment: 29 page
Alpha-invariants and purely log terminal blow-ups
We prove that the sum of the -invariants of two different Koll\'ar
components of a Kawamata log terminal singularity is less than .Comment: 12 page