18 research outputs found
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
Segments and Hilbert schemes of points
Using results obtained from the study of homogeneous ideals sharing the same
initial ideal with respect to some term order, we prove the singularity of the
point corresponding to a segment ideal with respect to the revlex term order in
the Hilbert scheme of points in . In this context, we look inside
properties of several types of "segment" ideals that we define and compare.
This study led us to focus our attention also to connections between the shape
of generators of Borel ideals and the related Hilbert polynomial, providing an
algorithm for computing all saturated Borel ideals with the given Hilbert
polynomial.Comment: 19 pages, 2 figures. Comments and suggestions are welcome
Oracle-supported drawing of the Groebner {\em escalier}
The aim of this note is to discuss the following quite queer Problem:
\noindent GIVEN \noindent i) the free non-commutative polynomial ring, {\Cal
P} := {\Bbb F}\langle X_1,\ldots,X_n\rangle {\em (public)}, \noindent ii) a
bilateral ideal {\em
(private)}, \noindent iii) a finite set
of elements of the ideal {\em (public)}, \noindent a noetherian
semigroup term-ordering {\rm (private)}, on the word semigroup {\Cal
T} := , \noindent COMPUTE \noindent --a finite subset
of the Gr\"obner basis of
w.r.t. s.t., for each its {\em normal form}
w.r.t. is zero, \noindent "by means of a finite number of queries to an
oracle", which, \noindent given a term \tau\in{\Cal T} returns its {\em
canonical form} \Can(\tau,{\sf I},\prec) w.r.t. the ideal and the
term-ordering . \qed This queer problem has been suggested to us by
Bulygin (2005) where a similar problem, but with stronger assumptions, is faced
in order to set up a chosen-cyphertext attack against the cryptographic system
proposed in Rai (2004)
The big mother of all dualities: Möller algorithm
Duality was introduced in Computer Algebra in 1982 by Möller and since that has been widely used. We give a survey of Möller algorithm and its applications, presenting a new one to the computation of canonical modules. "Its dual application" allow us to give answer to a question posed to us by Stette