6,777 research outputs found
When the positivity of the h-vector implies the Cohen-Macaulay property
We study relations between the Cohen-Macaulay property and the positivity of
-vectors, showing that these two conditions are equivalent for those locally
Cohen-Macaulay equidimensional closed projective subschemes , which are
close to a complete intersection (of the same codimension) in terms of the
difference between the degrees. More precisely, let
() be contained in , either of codimension two with
or of codimension with .
Over a field of characteristic 0, we prove that is arithmetically
Cohen-Macaulay if and only if its -vector is positive, improving results of
a previous work. We show that this equivalence holds also for space curves
with in every characteristic . Moreover, we
find other classes of subschemes for which the positivity of the -vector
implies the Cohen-Macaulay property and provide several examples.Comment: Main changes with respect the previuos version are in the title, the
abstract, the introduction and the bibliograph
Plucker-Clebsch formula in higher dimension
Let S\subset\Ps^r () be a nondegenerate, irreducible, smooth,
complex, projective surface of degree . Let be the number of
double points of a general projection of to \Ps^4. In the present paper
we prove that , with equality if and only if
is a rational scroll. Extensions to higher dimensions are discussed.Comment: 12 page
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
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