703 research outputs found
On the topology of a resolution of isolated singularities
Let be a complex projective variety of dimension with isolated
singularities, a resolution of singularities,
the exceptional locus. From Decomposition Theorem
one knows that the map
vanishes for . Assuming this vanishing, we give a short proof of
Decomposition Theorem for . A consequence is a short proof of the
Decomposition Theorem for in all cases where one can prove the vanishing
directly. This happens when either is a normal surface, or when is
the blowing-up of along with smooth and connected fibres,
or when admits a natural Gysin morphism. We prove that this last
condition is equivalent to say that the map vanishes for any , and that the pull-back
is injective. This provides a relationship between
Decomposition Theorem and Bivariant Theory.Comment: 18 page
N\'eron-Severi group of a general hypersurface
In this paper we extend the well known theorem of Angelo Lopez concerning the
Picard group of the general space projective surface containing a given smooth
projective curve, to the intermediate N\'eron-Severi group of a general
hypersurface in any smooth projective variety.Comment: 14 pages, to appear on Communications in Contemporary Mathematic
On a resolution of singularities with two strata
Let be a complex, irreducible, quasi-projective variety, and
a resolution of singularities of . Assume that the
singular locus of is smooth, that the induced map
is a smooth fibration
admitting a cohomology extension of the fiber, and that
has a negative normal bundle in . We
present a very short and explicit proof of the Decomposition Theorem for ,
providing a way to compute the intersection cohomology of by means of the
cohomology of and of . Our result
applies to special Schubert varieties with two strata, even if is
non-small. And to certain hypersurfaces of with one-dimensional
singular locus.Comment: 19 pages, no figure
Monodromy of a family of hypersurfaces
Let be an -dimensional irreducible smooth complex projective
variety embedded in a projective space. Let be a closed subscheme of ,
and be a positive integer such that is
generated by global sections. Fix an integer , and assume the
general divisor X \in |H^0(Y,\ic_{Z,Y}(d))| is smooth. Denote by
the quotient of by
the cohomology of and also by the cycle classes of the irreducible
components of dimension of . In the present paper we prove that the
monodromy representation on for the
family of smooth divisors X \in |H^0(Y,\ic_{Z,Y}(d))| is irreducible.Comment: 13 pages, to appear on Ann. Scient. Ec. Norm. Su
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