117 research outputs found
Strongly stable ideals and Hilbert polynomials
The \texttt{StronglyStableIdeals} package for \textit{Macaulay2} provides a
method to compute all saturated strongly stable ideals in a given polynomial
ring with a fixed Hilbert polynomial. A description of the main method and
auxiliary tools is given.Comment: Source code available as an ancillary file. Final versio
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
Computable Hilbert Schemes
In this PhD thesis we propose an algorithmic approach to the study of the
Hilbert scheme. Developing algorithmic methods, we also obtain general results
about Hilbert schemes. In Chapter 1 we discuss the equations defining the
Hilbert scheme as subscheme of a suitable Grassmannian and in Chapter 5 we
determine a new set of equations of degree lower than the degree of equations
known so far. In Chapter 2 we study the most important objects used to project
algorithmic techniques, namely Borel-fixed ideals. We determine an algorithm
computing all the saturated Borel-fixed ideals with Hilbert polynomial assigned
and we investigate their combinatorial properties. In Chapter 3 we show a new
type of flat deformations of Borel-fixed ideals which lead us to give a new
proof of the connectedness of the Hilbert scheme. In Chapter 4 we construct
families of ideals that generalize the notion of family of ideals sharing the
same initial ideal with respect to a fixed term ordering. Some of these
families correspond to open subsets of the Hilbert scheme and can be used to a
local study of the Hilbert scheme. In Chapter 6 we deal with the problem of the
connectedness of the Hilbert scheme of locally Cohen-Macaulay curves in the
projective 3-space. We show that one of the Hilbert scheme considered a "good"
candidate to be non-connected, is instead connected. Moreover there are three
appendices that present and explain how to use the implementations of the
algorithms proposed.Comment: This is the PhD thesis of the author. Most of the results appeared or
are going to appear in some paper. However the thesis contains more detailed
explanations, proofs and remarks and it can be used also as handbook for all
algorithms proposed and available at
http://www.personalweb.unito.it/paolo.lella/HSC/index.html . arXiv admin
note: text overlap with arXiv:1101.2866 by other author
A combinatorial description of finite O-sequences and aCM genera
The goal of this paper is to explicitly detect all the arithmetic genera of
arithmetically Cohen-Macaulay projective curves with a given degree . It is
well-known that the arithmetic genus of a curve can be easily deduced
from the -vector of the curve; in the case where is arithmetically
Cohen-Macaulay of degree , must belong to the range of integers
. We develop an algorithmic procedure that
allows one to avoid constructing most of the possible -vectors of . The
essential tools are a combinatorial description of the finite O-sequences of
multiplicity , and a sort of continuity result regarding the generation of
the genera. The efficiency of our method is supported by computational
evidence. As a consequence, we single out the minimal possible
Castelnuovo-Mumford regularity of a curve with Cohen-Macaulay postulation and
given degree and genus.Comment: Final versio
The maximum likelihood degree of Fermat hypersurfaces
We study the critical points of the likelihood function over the Fermat
hypersurface. This problem is related to one of the main problems in
statistical optimization: maximum likelihood estimation. The number of critical
points over a projective variety is a topological invariant of the variety and
is called maximum likelihood degree. We provide closed formulas for the maximum
likelihood degree of any Fermat curve in the projective plane and of Fermat
hypersurfaces of degree 2 in any projective space. Algorithmic methods to
compute the ML degree of a generic Fermat hypersurface are developed throughout
the paper. Such algorithms heavily exploit the symmetries of the varieties we
are considering. A computational comparison of the different methods and a list
of the maximum likelihood degrees of several Fermat hypersurfaces are available
in the last section.Comment: Final version. Accepted for publication on Journal of Algebraic
Statistic
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
On the functoriality of marked families
The application of methods of computational algebra has recently introduced new tools for the study of Hilbert schemes. The key idea is to define at families of ideals endowed with a scheme structure whose defining equations can be determined by algorithmic procedures. For this reason, several authors developed new methods, based on the combinatorial properties of Borel-fixed ideals, that allow associating to each ideal of this type a scheme , called a -marked scheme. In this paper, we provide a solid functorial foundation to marked schemes and show that the algorithmic procedures introduced in previous papers do not depend on the ring of coefficients. We prove that, for all strongly stable ideals , the marked schemes can be embedded in a Hilbert scheme as locally closed subschemes, and that they are open under suitable conditions on . Finally, we generalize Lederer's result about Gröbner strata of zero-dimensional ideals, proving that Gröbner strata of any ideals are locally closed subschemes of Hilbert schemes
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