When the positivity of the h-vector implies the Cohen-Macaulay property

We study relations between the Cohen-Macaulay property and the positivity of $h$-vectors, showing that these two conditions are equivalent for those locally Cohen-Macaulay equidimensional closed projective subschemes $X$, which are close to a complete intersection $Y$ (of the same codimension) in terms of the difference between the degrees. More precisely, let $X\subset \mathbb P^n_K$ ($n\geq 4$) be contained in $Y$, either of codimension two with $deg(Y)-deg(X)\leq 5$ or of codimension $\geq 3$ with $deg(Y)-deg(X)\leq 3$. Over a field $K$ of characteristic 0, we prove that $X$ is arithmetically Cohen-Macaulay if and only if its $h$-vector is positive, improving results of a previous work. We show that this equivalence holds also for space curves $C$ with $deg(Y)-deg(C)\leq 5$ in every characteristic $ch(K)\neq 2$. Moreover, we find other classes of subschemes for which the positivity of the $h$-vector implies the Cohen-Macaulay property and provide several examples.Comment: Main changes with respect the previuos version are in the title, the abstract, the introduction and the bibliograph

Functors of Liftings of Projective Schemes

A classical approach to investigate a closed projective scheme $W$ consists of considering a general hyperplane section of $W$, which inherits many properties of $W$. The inverse problem that consists in finding a scheme $W$ starting from a possible hyperplane section $Y$ is called a {\em lifting problem}, and every such scheme $W$ is called a {\em lifting} of $Y$. Investigations in this topic can produce methods to obtain schemes with specific properties. For example, any smooth point for $Y$ is smooth also for $W$. We characterize all the liftings of $Y$ 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

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.Comment: Final versio

The scheme of liftings and applications

We study the locus of the liftings of a homogeneous ideal $H$ in a polynomial ring over any field. We prove that this locus can be endowed with a structure of scheme $\mathrm L_H$ 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 $H$. We also provide an explicit isomorphism between the schemes corresponding to two different term orders. Our approach allows to embed $\mathrm L_H$ in a Hilbert scheme as a locally closed subscheme, and, over an infinite field, leads to find interesting topological properties, as for instance that $\mathrm L_H$ 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

Upgraded methods for the effective computation of marked schemes on a strongly stable ideal

Let $J\subset S=K[x_0,...,x_n]$ be a monomial strongly stable ideal. The collection \Mf(J) of the homogeneous polynomial ideals $I$, such that the monomials outside $J$ form a $K$-vector basis of $S/I$, is called a {\em $J$-marked family}. It can be endowed with a structure of affine scheme, called a {\em $J$-marked scheme}. For special ideals $J$, $J$-marked schemes provide an open cover of the Hilbert scheme \hilbp, where $p(t)$ is the Hilbert polynomial of $S/J$. Those ideals more suitable to this aim are the $m$-truncation ideals $\underline{J}_{\geq m}$ generated by the monomials of degree $\geq m$ in a saturated strongly stable monomial ideal $\underline{J}$. 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 $\underline{J}_{\geq m}$-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 $m$, we give a closed embedding \phi_m: \Mf(\underline{J}_{\geq m})\hookrightarrow \Mf(\underline{J}_{\geq m+1}), characterize those $\phi_m$ that are isomorphisms in terms of the monomial basis of $\underline{J}$, especially we characterize the minimum integer $m_0$ such that $\phi_m$ is an isomorphism for every $m\geq m_0$.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

Minimal Castelnuovo-Mumford regularity for a given Hilbert polynomial

Let $K$ be an algebraically closed field of null characteristic and $p(z)$ a Hilbert polynomial. We look for the minimal Castelnuovo-Mumford regularity $m_{p(z)}$ of closed subschemes of projective spaces over $K$ with Hilbert polynomial $p(z)$. Experimental evidences led us to consider the idea that $m_{p(z)}$ could be achieved by schemes having a suitable minimal Hilbert function. We give a constructive proof of this fact. Moreover, we are able to compute the minimal Castelnuovo-Mumford regularity $m_p(z)^{\varrho}$ of schemes with Hilbert polynomial $p(z)$ and given regularity $\varrho$ of the Hilbert function, and also the minimal Castelnuovo-Mumford regularity $m_u$ of schemes with Hilbert function $u$. These results find applications in the study of Hilbert schemes. They are obtained by means of minimal Hilbert functions and of two new constructive methods which are based on the notion of growth-height-lexicographic Borel set and called ideal graft and extended lifting.Comment: 21 pages. Comments are welcome. More concise version with a slight change in the title. A further revised version has been accepted for publication in Experimental Mathematic

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.Comment: 19 pages, 2 figures. Comments and suggestions are welcome