14 research outputs found
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
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
A general framework for Noetherian well ordered polynomial reductions
Polynomial reduction is one of the main tools in computational algebra with
innumerable applications in many areas, both pure and applied. Since many years
both the theory and an efficient design of the related algorithm have been
solidly established.
This paper presents a general definition of polynomial reduction structure,
studies its features and highlights the aspects needed in order to grant and to
efficiently test the main properties (noetherianity, confluence, ideal
membership).
The most significant aspect of this analysis is a negative reappraisal of the
role of the notion of term order which is usually considered a central and
crucial tool in the theory. In fact, as it was already established in the
computer science context in relation with termination of algorithms, most of
the properties can be obtained simply considering a well-founded ordering,
while the classical requirement that it be preserved by multiplication is
irrelevant.
The last part of the paper shows how the polynomial basis concepts present in
literature are interpreted in our language and their properties are
consequences of the general results established in the first part of the paper.Comment: 36 pages. New title and substantial improvements to the presentation
according to the comments of the reviewer
Flat families by strongly stable ideals and a generalization of Groebner bases
Let J be a strongly stable monomial ideal in S=K[x_1,...,x_n] and let Mf(J)
be the family of all homogeneous ideals I in S such that the set of all terms
outside J is a K-vector basis of the quotient S/I. We show that an ideal I
belongs to Mf(J) if and only if it is generated by a special set of
polynomials, the J-marked basis of I, that in some sense generalizes the notion
of reduced Groebner basis and its constructive capabilities. Indeed, although
not every J-marked basis is a Groebner basis with respect to some term order, a
sort of normal form modulo I (with the ideal I in Mf(J)) can be computed for
every homogeneous polynomial, so that a J-marked basis can be characterized by
a Buchberger-like criterion. Using J-marked bases, we prove that the family
Mf(J) can be endowed, in a very natural way, with a structure of affine scheme
that turns out to be homogeneous with respect to a non-standard grading and
flat in the origin (the point corresponding to J), thanks to properties of
J-marked bases analogous to those of Groebner bases about syzygies.Comment: This paper includes and extends the paper posed at arXiv:1005.0457.
Revised version for publication. Added reference
The close relation between border and Pommaret marked bases
Given a finite order ideal in the polynomial ring over a field , let be the border of
and the Pommaret basis of the ideal generated by the
terms outside . In the framework of reduction structures introduced
by Ceria, Mora, Roggero in 2019, we investigate relations among
-marked sets (resp. bases) and -marked sets (resp. bases).
We prove that a -marked set is a marked basis if and
only if the -marked set contained in is a
marked basis and generates the same ideal as . Using a functorial
description of these marked bases, as a byproduct we obtain that the affine
schemes respectively parameterizing -marked bases and
-marked bases are isomorphic. We are able to describe
this isomorphism as a projection that can be explicitly constructed without the
use of Gr\"obner elimination techniques. In particular, we obtain a
straightforward embedding of border schemes in smaller affine spaces.
Furthermore, we observe that Pommaret marked schemes give an open covering of
punctual Hilbert schemes. Several examples are given along all the paper.Comment: 17 pages; presentation improved, some references adde