arXiv subjects

Maria Grazia Marinari

Publications and source records attributed to Maria Grazia Marinari.

4 recordsLinked to original sources

arXiv2015

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 $d$. It is well-known that the arithmetic genus $g$ of a curve $C$ can be easily deduced from the $h$-vector of the curve; in the case where $C$ is arithmetically Cohen-Macaulay of degree $d$, $g$ must belong to the range of integers $\big\{0,\ldots,\binom{d-1}{2}\big\}$. We develop an algorithmic procedure that allows one to avoid constructing most of the possible $h$-vectors of $C$. The essential tools are a combinatorial description of the finite O-sequences of multiplicity $d$, 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.

math.AC↗

arXiv2010

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 ${\sf I}\subset {\Bbb F}\langle X_1,\ldots,X_n\rangle$ {\em (private)}, \noindent iii) a finite set $G := \{g_1,\ldots,g_l\}\subset{\sf I}$ of elements of the ideal ${\sf I}$ {\em (public)}, \noindent a noetherian semigroup term-ordering $\prec,$ {\rm (private)}, on the word semigroup ${\Cal T} := < X_1,\ldots,X_n>$, \noindent COMPUTE \noindent --a finite subset $H\subsetΓ({\sf I})$ of the Gröbner basis $Γ({\sf I})$ of ${\sf I}$ w.r.t. $\prec$ s.t., for each $g_i\in G$ its {\em normal form} $NF(g_i,H)$ w.r.t. $H$ is zero, \noindent "by means of a finite number of queries to an oracle", which, \noindent given a term $τ\in{\Cal T}$ returns its {\em canonical form} $\Can(τ,{\sf I},\prec)$ w.r.t. the ideal ${\sf I}$ and the term-ordering $\prec$. \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).

math.AC↗

arXiv2010

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 $\mathbb{P}^n$. 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.

math.AC↗

arXiv2006

Regularity bounds for curves by minimal generators and Hilbert function

Let $ρ_C$ be the regularity of the Hilbert function of a projective curve $C$ in $\mathbb P^n_K$ over an algebraically closed field $K$ and $α_1,...,α_{n-1}$ be minimal degrees for which there exists a complete intersection of type $(α_1,...,α_{n-1})$ containing the curve $C$. Then the Castelnuovo-Mumford regularity of $C$ is upper bounded by $\max\{ρ_C+1,α_1+...+α_{n-1}-(n-2)\}$. We study and, for space curves, refine the above bound providing several examples.

math.AG↗