146 research outputs found
Splitting type, global sections and Chern classes for torsion free sheaves on P^N
In this paper we compare a torsion free sheaf \FF on \PP^N and the free
vector bundle \oplus_{i=1}^n\OPN(b_i) having same rank and splitting type. We
show that the first one has always "less" global sections, while it has a
higher second Chern class. In both cases bounds for the difference are found in
terms of the maximal free subsheaves of \FF. As a consequence we obtain a
direct, easy and more general proof of the "Horrocks' splitting criterion",
also holding for torsion free sheaves, and lower bounds for the Chern classes
c_i(\FF(t)) of twists of \FF, only depending on some numerical invariants
of \FF. Especially, we prove for rank torsion free sheaves on \PP^N,
whose splitting type has no gap (i.e. for every
), the following formula for the discriminant:
\Delta(\FF):=2nc_2-(n-1)c_1^2\geq -{1/12}n^2(n^2-1) Finally in the case of
rank reflexive sheaves we obtain polynomial upper bounds for the absolute
value of the higher Chern classes c_3(\FF(t)), ..., c_n(\FF(t)), for the
dimension of the cohomology modules H^i\FF(t) and for the Castelnuovo-Mumford
regularity of \FF; these polynomial bounds only depend only on c_1(\FF),
c_2(\FF), the splitting type of \FF and .Comment: Final version, 15 page
On the dimension of the minimal vertex covers semigroup ring of an unmixed bipartite graph
In a paper in 2008, Herzog, Hibi and Ohsugi introduced and studied the
semigroup ring associated to the set of minimal vertex covers of an unmixed
bipartite graph. In this paper we relate the dimension of this semigroup ring
to the rank of the Boolean lattice associated to the graph.Comment: 6 pages, Pragmatic 2008, University of Catania (Italy); corrected
typo
A Borel open cover of the Hilbert scheme
Let be an admissible Hilbert polynomial in \PP^n of degree . The
Hilbert scheme \hilb^n_p(t) can be realized as a closed subscheme of a
suitable Grassmannian , hence it could be globally defined by
homogeneous equations in the Plucker coordinates of and covered by
open subsets given by the non-vanishing of a Plucker coordinate, each embedded
as a closed subscheme of the affine space , . However,
the number of Plucker coordinates is so large that effective computations
in this setting are practically impossible. In this paper, taking advantage of
the symmetries of \hilb^n_p(t), we exhibit a new open cover, consisting of
marked schemes over Borel-fixed ideals, whose number is significantly smaller
than . Exploiting the properties of marked schemes, we prove that these open
subsets are defined by equations of degree in their natural
embedding in \Af^D. Furthermore we find new embeddings in affine spaces of
far lower dimension than , and characterize those that are still defined by
equations of degree . The proofs are constructive and use a
polynomial reduction process, similar to the one for Grobner bases, but are
term order free. In this new setting, we can achieve explicit computations in
many non-trivial cases.Comment: 17 pages. This version contains and extends the first part of version
2 (arXiv:0909.2184v2[math.AG]). A new extended version of the second part,
with some new results, is posed at arxiv:1110.0698v3[math.AC]. The title is
slightly changed. Final version accepted for publicatio
Functors of Liftings of Projective Schemes
A classical approach to investigate a closed projective scheme consists
of considering a general hyperplane section of , which inherits many
properties of . The inverse problem that consists in finding a scheme
starting from a possible hyperplane section is called a {\em lifting
problem}, and every such scheme is called a {\em lifting} of .
Investigations in this topic can produce methods to obtain schemes with
specific properties. For example, any smooth point for is smooth also for
.
We characterize all the liftings of with a given Hilbert polynomial by a
parameter scheme that is obtained by gluing suitable affine open subschemes in
a Hilbert scheme and is described through the functor it represents. We use
constructive methods from Gr\"obner and marked bases theories. Furthermore, by
classical tools we obtain an analogous result for equidimensional liftings.
Examples of explicit computations are provided.Comment: 25 pages. Final version. Ancillary files available at
http://wpage.unina.it/cioffifr/MaterialeCoCoALiftingGeometric
The Euler characteristic as a polynomial in the Chern classes
In this paper we obtain some explicit expressions for the Euler
characteristic of a rank n coherent sheaf F on P^N and of its twists F(t) as
polynomials in the Chern classes c_i(F), also giving algorithms for the
computation. The employed methods use techniques of umbral calculus involving
symmetric functions and Stirling numbers.Comment: 12 page
The cones of Hilbert functions of squarefree modules
In this paper, we study different generalizations of the notion of
squarefreeness for ideals to the more general case of modules. We describe the
cones of Hilbert functions for squarefree modules in general and those
generated in degree zero. We give their extremal rays and defining
inequalities. For squarefree modules generated in degree zero, we compare the
defining inequalities of that cone with the classical Kruskal-Katona bound,
also asymptotically.Comment: 17 pages, 2 figures. This paper was produced during Pragmatic 201
Upgraded methods for the effective computation of marked schemes on a strongly stable ideal
Let be a monomial strongly stable ideal. The
collection \Mf(J) of the homogeneous polynomial ideals , such that the
monomials outside form a -vector basis of , is called a {\em
-marked family}. It can be endowed with a structure of affine scheme, called
a {\em -marked scheme}. For special ideals , -marked schemes provide
an open cover of the Hilbert scheme \hilbp, where is the Hilbert
polynomial of . Those ideals more suitable to this aim are the
-truncation ideals generated by the monomials of
degree in a saturated strongly stable monomial ideal .
Exploiting a characterization of the ideals in \Mf(\underline{J}_{\geq m}) in
terms of a Buchberger-like criterion, we compute the equations defining the
-marked scheme by a new reduction relation, called {\em
superminimal reduction}, and obtain an embedding of \Mf(\underline{J}_{\geq
m}) in an affine space of low dimension. In this setting, explicit
computations are achievable in many non-trivial cases. Moreover, for every ,
we give a closed embedding \phi_m: \Mf(\underline{J}_{\geq m})\hookrightarrow
\Mf(\underline{J}_{\geq m+1}), characterize those that are
isomorphisms in terms of the monomial basis of , especially we
characterize the minimum integer such that is an isomorphism for
every .Comment: 28 pages; this paper contains and extends the second part of the
paper posed at arXiv:0909.2184v2[math.AG]; sections are now reorganized and
the general presentation of the paper is improved. Final version accepted for
publicatio
The scheme of liftings and applications
We study the locus of the liftings of a homogeneous ideal in a polynomial
ring over any field. We prove that this locus can be endowed with a structure
of scheme by applying the constructive methods of Gr\"obner
bases, for any given term order. Indeed, this structure does not depend on the
term order, since it can be defined as the scheme representing the functor of
liftings of . We also provide an explicit isomorphism between the schemes
corresponding to two different term orders.
Our approach allows to embed in a Hilbert scheme as a locally
closed subscheme, and, over an infinite field, leads to find interesting
topological properties, as for instance that is connected and
that its locus of radical liftings is open. Moreover, we show that every ideal
defining an arithmetically Cohen-Macaulay scheme of codimension two has a
radical lifting, giving in particular an answer to an open question posed by L.
G. Roberts in 1989.Comment: the presentation of the results has been improved, new section
(Section 6 of this version) concerning the torus action on the scheme of
liftings, more detailed proofs in Section 7 of this version (Section 6 in the
previous version), new example added (Example 8.5 of this version
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